sin(x)^4-cos(x)^2*sin(x)^ ... +cos(x)^4-sin(x)^6еслиx=3 (упростите выражение)

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Решение

Вы ввели [src]
   4         2       2         4         6   
sin (x) - cos (x)*sin (x) + cos (x) - sin (x)
sin6(x)+sin4(x)sin2(x)cos2(x)+cos4(x)- \sin^{6}{\left(x \right)} + \sin^{4}{\left(x \right)} - \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} + \cos^{4}{\left(x \right)}
Подстановка условия [src]
sin(x)^4 - cos(x)^2*sin(x)^2 + cos(x)^4 - sin(x)^6 при x = 3
подставляем
   4         2       2         4         6   
sin (x) - cos (x)*sin (x) + cos (x) - sin (x)
sin6(x)+sin4(x)sin2(x)cos2(x)+cos4(x)- \sin^{6}{\left(x \right)} + \sin^{4}{\left(x \right)} - \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} + \cos^{4}{\left(x \right)}
   6   
cos (x)
cos6(x)\cos^{6}{\left(x \right)}
переменные
x = 3
x=3x = 3
   6     
cos ((3))
cos6((3))\cos^{6}{\left((3) \right)}
   6   
cos (3)
cos6(3)\cos^{6}{\left(3 \right)}
Степени [src]
                                                                        2                  
                                                          / I*x    -I*x\                  2
              4                   4                   6   |e      e    |  /   -I*x    I*x\ 
/ I*x    -I*x\    /   -I*x    I*x\    /   -I*x    I*x\    |---- + -----| *\- e     + e   / 
|e      e    |    \- e     + e   /    \- e     + e   /    \ 2       2  /                   
|---- + -----|  + ----------------- + ----------------- + ---------------------------------
\ 2       2  /            16                  64                          4                
(eix2+eix2)4+(eix2+eix2)2(eixeix)24+(eixeix)664+(eixeix)416\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{4} + \frac{\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2} \left(e^{i x} - e^{- i x}\right)^{2}}{4} + \frac{\left(e^{i x} - e^{- i x}\right)^{6}}{64} + \frac{\left(e^{i x} - e^{- i x}\right)^{4}}{16}
Численный ответ [src]
cos(x)^4 + sin(x)^4 - sin(x)^6 - cos(x)^2*sin(x)^2
Рациональный знаменатель [src]
   4         4         6         2       2   
cos (x) + sin (x) - sin (x) - cos (x)*sin (x)
sin6(x)+sin4(x)sin2(x)cos2(x)+cos4(x)- \sin^{6}{\left (x \right )} + \sin^{4}{\left (x \right )} - \sin^{2}{\left (x \right )} \cos^{2}{\left (x \right )} + \cos^{4}{\left (x \right )}
Объединение рациональных выражений [src]
   4         6         2    /   2         2   \
cos (x) - sin (x) + sin (x)*\sin (x) - cos (x)/
(sin2(x)cos2(x))sin2(x)sin6(x)+cos4(x)\left(\sin^{2}{\left (x \right )} - \cos^{2}{\left (x \right )}\right) \sin^{2}{\left (x \right )} - \sin^{6}{\left (x \right )} + \cos^{4}{\left (x \right )}
Общее упрощение [src]
   6   
cos (x)
cos6(x)\cos^{6}{\left(x \right)}
Собрать выражение [src]
5    cos(6*x)   3*cos(4*x)   15*cos(2*x)
-- + -------- + ---------- + -----------
16      32          16            32    
1532cos(2x)+316cos(4x)+132cos(6x)+516\frac{15}{32} \cos{\left (2 x \right )} + \frac{3}{16} \cos{\left (4 x \right )} + \frac{1}{32} \cos{\left (6 x \right )} + \frac{5}{16}
   4         4         6         2       2   
cos (x) + sin (x) - sin (x) - cos (x)*sin (x)
sin6(x)+sin4(x)sin2(x)cos2(x)+cos4(x)- \sin^{6}{\left (x \right )} + \sin^{4}{\left (x \right )} - \sin^{2}{\left (x \right )} \cos^{2}{\left (x \right )} + \cos^{4}{\left (x \right )}
Общий знаменатель [src]
   4         4         6         2       2   
cos (x) + sin (x) - sin (x) - cos (x)*sin (x)
sin6(x)+sin4(x)sin2(x)cos2(x)+cos4(x)- \sin^{6}{\left (x \right )} + \sin^{4}{\left (x \right )} - \sin^{2}{\left (x \right )} \cos^{2}{\left (x \right )} + \cos^{4}{\left (x \right )}
Тригонометрическая часть [src]
     1      
------------
   6/pi    \
csc |-- - x|
    \2     /
1csc6(x+π2)\frac{1}{\csc^{6}{\left(- x + \frac{\pi}{2} \right)}}
                                                                                                                                                                                                         //        1           for And(im(x) = 0, x mod 2*pi = 0)\
                                                                                                                                                                                                         ||                                                      |
  //   0     for And(im(x) = 0, x mod pi = 0)\   //   0     for And(im(x) = 0, x mod pi = 0)\ //   1     for And(im(x) = 0, x mod 2*pi = 0)\   //        0           for And(im(x) = 0, x mod pi = 0)\   ||      4       8/x\                                    |
  ||                                         |   ||                                         | ||                                           |   ||                                                    |   ||16*cos (x)*sin |-|                                    |
- |<   6                                     | - |<   2                                     |*|<   2                                       | + |<      4/x\    4/x\                                  | + |<               \2/                                    |
  ||sin (x)             otherwise            |   ||sin (x)             otherwise            | ||cos (x)              otherwise             |   ||16*cos |-|*sin |-|             otherwise            |   ||------------------              otherwise             |
  \\                                         /   \\                                         / \\                                           /   \\       \2/     \2/                                  /   ||               4                                      |
                                                                                                                                                                                                         ||  (-1 + cos(x))                                       |
                                                                                                                                                                                                         \\                                                      /
({0forim(x)=0xmodπ=016sin4(x2)cos4(x2)otherwise)(({0forim(x)=0xmodπ=0sin2(x)otherwise)({1forim(x)=0xmod2π=0cos2(x)otherwise))({0forim(x)=0xmodπ=0sin6(x)otherwise)+({1forim(x)=0xmod2π=016sin8(x2)cos4(x)(cos(x)1)4otherwise)\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\16 \sin^{4}{\left(\frac{x}{2} \right)} \cos^{4}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{16 \sin^{8}{\left(\frac{x}{2} \right)} \cos^{4}{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right)^{4}} & \text{otherwise} \end{cases}\right)
             6
/       2/x\\ 
|1 - tan |-|| 
\        \2// 
--------------
             6
/       2/x\\ 
|1 + tan |-|| 
\        \2// 
(1tan2(x2))6(tan2(x2)+1)6\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{6}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}}
                                                                                                                                                                                                         //           1             for And(im(x) = 0, x mod 2*pi = 0)\
  //   0     for And(im(x) = 0, x mod pi = 0)\   //   0     for And(im(x) = 0, x mod pi = 0)\ //   1     for And(im(x) = 0, x mod 2*pi = 0)\   //        0           for And(im(x) = 0, x mod pi = 0)\   ||                                                           |
  ||                                         |   ||                                         | ||                                           |   ||                                                    |   ||              4                                            |
- |<   6                                     | - |<   2                                     |*|<   2                                       | + |<      4/x\    8/x\                                  | + |
({0forim(x)=0xmodπ=016sin8(x2)cot4(x2)otherwise)(({0forim(x)=0xmodπ=0sin2(x)otherwise)({1forim(x)=0xmod2π=0cos2(x)otherwise))({0forim(x)=0xmodπ=0sin6(x)otherwise)+({1forim(x)=0xmod2π=0(cot2(x2)1)4sin8(x2)otherwise)\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\16 \sin^{8}{\left(\frac{x}{2} \right)} \cot^{4}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right)
   1           1              1                  1          
------- + ------------ - ------------ - --------------------
   4         4/    pi\      6/    pi\      2       2/    pi\
sec (x)   sec |x - --|   sec |x - --|   sec (x)*sec |x - --|
              \    2 /       \    2 /               \    2 /
1sec4(xπ2)1sec6(xπ2)1sec2(x)sec2(xπ2)+1sec4(x)\frac{1}{\sec^{4}{\left(x - \frac{\pi}{2} \right)}} - \frac{1}{\sec^{6}{\left(x - \frac{\pi}{2} \right)}} - \frac{1}{\sec^{2}{\left(x \right)} \sec^{2}{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec^{4}{\left(x \right)}}
                                                                                                                                                                                                                //         1            for And(im(x) = 0, x mod 2*pi = 0)\
                                                                                                                                                                                                                ||                                                        |
                                                                                                                                                                                                                ||                   4                                    |
                                                                                                                                                                                                                ||/          2/x\   \                                     |
  //   0     for And(im(x) = 0, x mod pi = 0)\   //   0     for And(im(x) = 0, x mod pi = 0)\ //     1        for And(im(x) = 0, x mod 2*pi = 0)\   //         0            for And(im(x) = 0, x mod pi = 0)\   |||       csc |-|   |                                     |
  ||                                         |   ||                                         | ||                                                |   ||                                                      |   |||           \2/   |                                     |
  ||   1                                     |   ||   1                                     | ||     1                                          |   ||         16                                           |   |||-1 + ------------|                                     |
- |<-------             otherwise            | - |<-------             otherwise            |*|<------------              otherwise             | + |<--------------------             otherwise            | + |<|        2/pi   x\|                                     |
  ||   6                                     |   ||   2                                     | ||   2/pi    \                                    |   ||   4/x\    4/pi   x\                                  |   |||     csc |-- - -||                                     |
  ||csc (x)                                  |   ||csc (x)                                  | ||csc |-- - x|                                    |   ||csc |-|*csc |-- - -|                                  |   ||\         \2    2//                                     |
  \\                                         /   \\                                         / \\    \2     /                                    /   \\    \2/     \2    2/                                  /   ||--------------------              otherwise             |
                                                                                                                                                                                                                ||         8/x\                                           |
                                                                                                                                                                                                                ||      csc |-|                                           |
                                                                                                                                                                                                                ||          \2/                                           |
                                                                                                                                                                                                                \\                                                        /
({0forim(x)=0xmodπ=016csc4(x2)csc4(x2+π2)otherwise)({0forim(x)=0xmodπ=01csc6(x)otherwise)(({0forim(x)=0xmodπ=01csc2(x)otherwise)({1forim(x)=0xmod2π=01csc2(x+π2)otherwise))+({1forim(x)=0xmod2π=0(csc2(x2)csc2(x2+π2)1)4csc8(x2)otherwise)\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{16}{\csc^{4}{\left(\frac{x}{2} \right)} \csc^{4}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{1}{\csc^{6}{\left(x \right)}} & \text{otherwise} \end{cases}\right) - \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right)^{4}}{\csc^{8}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right)
   4         4/    pi\      6         2       2/    pi\
sin (x) + sin |x + --| - sin (x) - sin (x)*sin |x + --|
              \    2 /                         \    2 /
sin6(x)+sin4(x)sin2(x)sin2(x+π2)+sin4(x+π2)- \sin^{6}{\left(x \right)} + \sin^{4}{\left(x \right)} - \sin^{2}{\left(x \right)} \sin^{2}{\left(x + \frac{\pi}{2} \right)} + \sin^{4}{\left(x + \frac{\pi}{2} \right)}
        6/x\        6/pi   x\        8/pi   x\         4/x\         8/x\         6/x\    6/pi   x\
- 68*sin |-| - 8*sin |-- + -| + 9*sin |-- + -| + 30*sin |-| + 39*sin |-| - 64*sin |-|*sin |-- + -|
         \2/         \2    2/         \2    2/          \2/          \2/          \2/     \2    2/
39sin8(x2)64sin6(x2)sin6(x2+π2)68sin6(x2)+30sin4(x2)+9sin8(x2+π2)8sin6(x2+π2)39 \sin^{8}{\left(\frac{x}{2} \right)} - 64 \sin^{6}{\left(\frac{x}{2} \right)} \sin^{6}{\left(\frac{x}{2} + \frac{\pi}{2} \right)} - 68 \sin^{6}{\left(\frac{x}{2} \right)} + 30 \sin^{4}{\left(\frac{x}{2} \right)} + 9 \sin^{8}{\left(\frac{x}{2} + \frac{\pi}{2} \right)} - 8 \sin^{6}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}
   1   
-------
   6   
sec (x)
1sec6(x)\frac{1}{\sec^{6}{\left(x \right)}}
/           1              for And(im(x) = 0, x mod 2*pi = 0)
|                                                            
|              6                                             
{1forim(x)=0xmod2π=0(cot2(x2)1)6sin12(x2)otherwise\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{6} \sin^{12}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}
                                                                                                                                                                                                   //            1              for And(im(x) = 0, x mod 2*pi = 0)\
                                                                                                                                                                                                   ||                                                             |
  //   0     for And(im(x) = 0, x mod pi = 0)\   //   0     for And(im(x) = 0, x mod pi = 0)\ //     1        for And(im(x) = 0, x mod 2*pi = 0)\   //   0     for And(im(x) = 0, x mod pi = 0)\   ||                4                                            |
  ||                                         |   ||                                         | ||                                                |   ||                                         |   ||/         2    \                                             |
- |<   6                                     | - |<   2                                     |*|<   2/    pi\                                    | + |<   4                                     | + |<|      sin (x) |     8/x\                                    |
  ||sin (x)             otherwise            |   ||sin (x)             otherwise            | ||sin |x + --|              otherwise             |   ||sin (x)             otherwise            |   |||-1 + ---------| *sin |-|              otherwise             |
  \\                                         /   \\                                         / \\    \    2 /                                    /   \\                                         /   |||          4/x\|      \2/                                    |
                                                                                                                                                                                                   |||     4*sin |-||                                             |
                                                                                                                                                                                                   \\\           \2//                                             /
(({0forim(x)=0xmodπ=0sin2(x)otherwise)({1forim(x)=0xmod2π=0sin2(x+π2)otherwise))+({0forim(x)=0xmodπ=0sin4(x)otherwise)({0forim(x)=0xmodπ=0sin6(x)otherwise)+({1forim(x)=0xmod2π=0(1+sin2(x)4sin4(x2))4sin8(x2)otherwise)\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\sin^{2}{\left(x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{4}{\left(x \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right)
                                                                                                                                                                                              //            1              for And(im(x) = 0, x mod 2*pi = 0)\
                                                                                                                                                                                              ||                                                             |
  //   0     for And(im(x) = 0, x mod pi = 0)\   //   0     for And(im(x) = 0, x mod pi = 0)\ //   1     for And(im(x) = 0, x mod 2*pi = 0)\   //   0     for And(im(x) = 0, x mod pi = 0)\   ||                4                                            |
  ||                                         |   ||                                         | ||                                           |   ||                                         |   ||/         2    \                                             |
- |<   6                                     | - |<   2                                     |*|<   2                                       | + |<   4                                     | + |<|      sin (x) |     8/x\                                    |
  ||sin (x)             otherwise            |   ||sin (x)             otherwise            | ||cos (x)              otherwise             |   ||sin (x)             otherwise            |   |||-1 + ---------| *sin |-|              otherwise             |
  \\                                         /   \\                                         / \\                                           /   \\                                         /   |||          4/x\|      \2/                                    |
                                                                                                                                                                                              |||     4*sin |-||                                             |
                                                                                                                                                                                              \\\           \2//                                             /
(({0forim(x)=0xmodπ=0sin2(x)otherwise)({1forim(x)=0xmod2π=0cos2(x)otherwise))+({0forim(x)=0xmodπ=0sin4(x)otherwise)({0forim(x)=0xmodπ=0sin6(x)otherwise)+({1forim(x)=0xmod2π=0(1+sin2(x)4sin4(x2))4sin8(x2)otherwise)\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{4}{\left(x \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right)
   1         1         1             1       
------- + ------- - ------- - ---------------
   4         4         6         2       2   
csc (x)   sec (x)   csc (x)   csc (x)*sec (x)
1sec4(x)1csc2(x)sec2(x)+1csc4(x)1csc6(x)\frac{1}{\sec^{4}{\left(x \right)}} - \frac{1}{\csc^{2}{\left(x \right)} \sec^{2}{\left(x \right)}} + \frac{1}{\csc^{4}{\left(x \right)}} - \frac{1}{\csc^{6}{\left(x \right)}}
   4         4/    pi\      6/    pi\      2       2/    pi\
cos (x) + cos |x - --| - cos |x - --| - cos (x)*cos |x - --|
              \    2 /       \    2 /               \    2 /
cos4(x)cos2(x)cos2(xπ2)cos6(xπ2)+cos4(xπ2)\cos^{4}{\left(x \right)} - \cos^{2}{\left(x \right)} \cos^{2}{\left(x - \frac{\pi}{2} \right)} - \cos^{6}{\left(x - \frac{\pi}{2} \right)} + \cos^{4}{\left(x - \frac{\pi}{2} \right)}
     //                       /           x           \\     //                        /           x             \\     //                        /           x             \\      //                       /           x           \\      //                       /           x           \\      //                       /           x           \\ //                        /           x             \\
     ||      0         for And|im(x) = 0, - mod pi = 0||     ||       1         for And|im(x) = 0, - mod 2*pi = 0||     ||       1         for And|im(x) = 0, - mod 2*pi = 0||      ||      0         for And|im(x) = 0, - mod pi = 0||      ||      0         for And|im(x) = 0, - mod pi = 0||      ||      0         for And|im(x) = 0, - mod pi = 0|| ||       1         for And|im(x) = 0, - mod 2*pi = 0||
     ||                       \           2           /|     ||                        \           2             /|     ||                        \           2             /|      ||                       \           2           /|      ||                       \           2           /|      ||                       \           2           /| ||                        \           2             /|
     ||                                                |     ||                                                   |     ||                                                   |      ||                                                |      ||                                                |      ||                                                | ||                                                   |
     ||        6/x\                                    |     ||              6                                    |     ||              8                                    |      ||        4/x\                                    |      ||        8/x\                                    |      ||        6/x\                                    | ||              6                                    |
     ||  64*cot |-|                                    |     ||/        2/x\\                                     |     ||/        2/x\\                                     |      ||  16*cot |-|                                    |      || 256*cot |-|                                    |      ||  64*cot |-|                                    | ||/        2/x\\                                     |
- 68*|<         \4/                                    | - 8*|<|-1 + cot |-||                                     | + 9*|<|-1 + cot |-||                                     | + 30*|<         \4/                                    | + 39*|<         \4/                                    | - 64*|<         \4/                                    |*|<|-1 + cot |-||                                     |
     ||--------------             otherwise            |     ||\         \4//                                     |     ||\         \4//                                     |      ||--------------             otherwise            |      ||--------------             otherwise            |      ||--------------             otherwise            | ||\         \4//                                     |
     ||             6                                  |     ||---------------              otherwise             |     ||---------------              otherwise             |      ||             4                                  |      ||             8                                  |      ||             6                                  | ||---------------              otherwise             |
     ||/       2/x\\                                   |     ||              6                                    |     ||              8                                    |      ||/       2/x\\                                   |      ||/       2/x\\                                   |      ||/       2/x\\                                   | ||              6                                    |
     |||1 + cot |-||                                   |     || /       2/x\\                                     |     || /       2/x\\                                     |      |||1 + cot |-||                                   |      |||1 + cot |-||                                   |      |||1 + cot |-||                                   | || /       2/x\\                                     |
     ||\        \4//                                   |     || |1 + cot |-||                                     |     || |1 + cot |-||                                     |      ||\        \4//                                   |      ||\        \4//                                   |      ||\        \4//                                   | || |1 + cot |-||                                     |
     \\                                                /     \\ \        \4//                                     /     \\ \        \4//                                     /      \\                                                /      \\                                                /      \\                                                / \\ \        \4//                                     /
(39({0forim(x)=0x2modπ=0256cot8(x4)(cot2(x4)+1)8otherwise))(64({0forim(x)=0x2modπ=064cot6(x4)(cot2(x4)+1)6otherwise)({1forim(x)=0x2mod2π=0(cot2(x4)1)6(cot2(x4)+1)6otherwise))(68({0forim(x)=0x2modπ=064cot6(x4)(cot2(x4)+1)6otherwise))+(30({0forim(x)=0x2modπ=016cot4(x4)(cot2(x4)+1)4otherwise))(8({1forim(x)=0x2mod2π=0(cot2(x4)1)6(cot2(x4)+1)6otherwise))+(9({1forim(x)=0x2mod2π=0(cot2(x4)1)8(cot2(x4)+1)8otherwise))\left(39 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod \pi = 0 \\\frac{256 \cot^{8}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right)\right) - \left(64 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod \pi = 0 \\\frac{64 \cot^{6}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{6}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right)\right) - \left(68 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod \pi = 0 \\\frac{64 \cot^{6}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right)\right) + \left(30 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod \pi = 0 \\\frac{16 \cot^{4}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)\right) - \left(8 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{6}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right)\right) + \left(9 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{8}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right)\right)
                                                                                                                                                                                                                     //         1            for And(im(x) = 0, x mod 2*pi = 0)\
                                                                                                                                                                                                                     ||                                                        |
                                                                                                                                                                                                                     ||                   4                                    |
                                                                                                                                                                                                                     ||/        2/x   pi\\                                     |
  //     0        for And(im(x) = 0, x mod pi = 0)\   //     0        for And(im(x) = 0, x mod pi = 0)\ //   1     for And(im(x) = 0, x mod 2*pi = 0)\   //         0            for And(im(x) = 0, x mod pi = 0)\   |||     sec |- - --||                                     |
  ||                                              |   ||                                              | ||                                           |   ||                                                      |   |||         \2   2 /|                                     |
  ||     1                                        |   ||     1                                        | ||   1                                       |   ||         16                                           |   |||-1 + ------------|                                     |
- |<------------             otherwise            | - |<------------             otherwise            |*|<-------              otherwise             | + |<--------------------             otherwise            | + |<|          2/x\   |                                     |
  ||   6/    pi\                                  |   ||   2/    pi\                                  | ||   2                                       |   ||   4/x\    4/x   pi\                                  |   |||       sec |-|   |                                     |
  ||sec |x - --|                                  |   ||sec |x - --|                                  | ||sec (x)                                    |   ||sec |-|*sec |- - --|                                  |   ||\           \2/   /                                     |
  \\    \    2 /                                  /   \\    \    2 /                                  / \\                                           /   \\    \2/     \2   2 /                                  /   ||--------------------              otherwise             |
                                                                                                                                                                                                                     ||       8/x   pi\                                        |
                                                                                                                                                                                                                     ||    sec |- - --|                                        |
                                                                                                                                                                                                                     ||        \2   2 /                                        |
                                                                                                                                                                                                                     \\                                                        /
({0forim(x)=0xmodπ=016sec4(x2)sec4(x2π2)otherwise)({0forim(x)=0xmodπ=01sec6(xπ2)otherwise)(({0forim(x)=0xmodπ=01sec2(xπ2)otherwise)({1forim(x)=0xmod2π=01sec2(x)otherwise))+({1forim(x)=0xmod2π=0(1+sec2(x2π2)sec2(x2))4sec8(x2π2)otherwise)\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{16}{\sec^{4}{\left(\frac{x}{2} \right)} \sec^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{1}{\sec^{6}{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{4}}{\sec^{8}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)
                                         6                           8                                                                            6                 
          12/x\    6/x\     /       2/x\\     12/x\     /       2/x\\     16/x\          8/x\    4/x\           16/x\    8/x\        /       2/x\\     24/x\    6/x\
- 4352*cos  |-|*tan |-| - 8*|1 - tan |-|| *cos  |-| + 9*|1 - tan |-|| *cos  |-| + 480*cos |-|*tan |-| + 9984*cos  |-|*tan |-| - 4096*|1 - tan |-|| *cos  |-|*tan |-|
            \4/     \4/     \        \4//       \4/     \        \4//       \4/           \4/     \4/             \4/     \4/        \        \4//       \4/     \4/
9(1tan2(x4))8cos16(x4)4096(1tan2(x4))6cos24(x4)tan6(x4)8(1tan2(x4))6cos12(x4)+9984cos16(x4)tan8(x4)4352cos12(x4)tan6(x4)+480cos8(x4)tan4(x4)9 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{8} \cos^{16}{\left(\frac{x}{4} \right)} - 4096 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{6} \cos^{24}{\left(\frac{x}{4} \right)} \tan^{6}{\left(\frac{x}{4} \right)} - 8 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{6} \cos^{12}{\left(\frac{x}{4} \right)} + 9984 \cos^{16}{\left(\frac{x}{4} \right)} \tan^{8}{\left(\frac{x}{4} \right)} - 4352 \cos^{12}{\left(\frac{x}{4} \right)} \tan^{6}{\left(\frac{x}{4} \right)} + 480 \cos^{8}{\left(\frac{x}{4} \right)} \tan^{4}{\left(\frac{x}{4} \right)}
        6/x   pi\        6/x\        8/x\         4/x   pi\         8/x   pi\         6/x\    6/x   pi\
- 68*cos |- - --| - 8*cos |-| + 9*cos |-| + 30*cos |- - --| + 39*cos |- - --| - 64*cos |-|*cos |- - --|
         \2   2 /         \2/         \2/          \2   2 /          \2   2 /          \2/     \2   2 /
9cos8(x2)64cos6(x2)cos6(x2π2)8cos6(x2)+39cos8(x2π2)68cos6(x2π2)+30cos4(x2π2)9 \cos^{8}{\left(\frac{x}{2} \right)} - 64 \cos^{6}{\left(\frac{x}{2} \right)} \cos^{6}{\left(\frac{x}{2} - \frac{\pi}{2} \right)} - 8 \cos^{6}{\left(\frac{x}{2} \right)} + 39 \cos^{8}{\left(\frac{x}{2} - \frac{\pi}{2} \right)} - 68 \cos^{6}{\left(\frac{x}{2} - \frac{\pi}{2} \right)} + 30 \cos^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}
                                                                                                                                                                                                                                  //             1               for And(im(x) = 0, x mod 2*pi = 0)\
  //      0         for And(im(x) = 0, x mod pi = 0)\   //      0         for And(im(x) = 0, x mod pi = 0)\ //      1         for And(im(x) = 0, x mod 2*pi = 0)\   //          0             for And(im(x) = 0, x mod pi = 0)\   ||                                                               |
  ||                                                |   ||                                                | ||                                                  |   ||                                                        |   ||                  4                                            |
  ||        6/x\                                    |   ||       2/x\                                     | ||             2                                    |   ||             8/x\                                       |   ||    /        1   \     8/x\                                    |
  ||  64*tan |-|                                    |   ||  4*tan |-|                                     | ||/       2/x\\                                     |   ||     4096*tan |-|                                       |   ||256*|-1 + -------| *tan |-|                                    |
  ||         \2/                                    |   ||        \2/                                     | |||1 - tan |-||                                     |   ||              \4/                                       |   ||    |        2/x\|      \4/                                    |
- |<--------------             otherwise            | - |<--------------             otherwise            |*|<\        \2//                                     | + |<----------------------             otherwise            | + |<    |     tan |-||                                             |
  ||             6                                  |   ||             2                                  | ||--------------              otherwise             |   ||             8                                          |   ||    \         \2//                                             |
  ||/       2/x\\                                   |   ||/       2/x\\                                   | ||             2                                    |   ||/       2/x\\     4/x\                                  |   ||---------------------------              otherwise             |
  |||1 + tan |-||                                   |   |||1 + tan |-||                                   | ||/       2/x\\                                     |   |||1 + tan |-|| *tan |-|                                  |   ||                    8                                          |
  ||\        \2//                                   |   ||\        \2//                                   | |||1 + tan |-||                                     |   ||\        \4//      \2/                                  |   ||       /       2/x\\                                           |
  \\                                                /   \\                                                / \\\        \2//                                     /   \\                                                        /   ||       |1 + tan |-||                                           |
                                                                                                                                                                                                                                  \\       \        \4//                                           /
({0forim(x)=0xmodπ=064tan6(x2)(tan2(x2)+1)6otherwise)(({0forim(x)=0xmodπ=04tan2(x2)(tan2(x2)+1)2otherwise)({1forim(x)=0xmod2π=0(1tan2(x2))2(tan2(x2)+1)2otherwise))+({0forim(x)=0xmodπ=04096tan8(x4)(tan2(x4)+1)8tan4(x2)otherwise)+({1forim(x)=0xmod2π=0256(1+1tan2(x2))4tan8(x4)(tan2(x4)+1)8otherwise)\left(- \begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{64 \tan^{6}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right) - \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{4096 \tan^{8}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8} \tan^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{256 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \tan^{8}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right)
   1           1            1               1          
------- + ------------ - ------- - --------------------
   4         4/pi    \      6         2       2/pi    \
csc (x)   csc |-- - x|   csc (x)   csc (x)*csc |-- - x|
              \2     /                         \2     /
1csc4(x+π2)1csc2(x)csc2(x+π2)+1csc4(x)1csc6(x)\frac{1}{\csc^{4}{\left(- x + \frac{\pi}{2} \right)}} - \frac{1}{\csc^{2}{\left(x \right)} \csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{4}{\left(x \right)}} - \frac{1}{\csc^{6}{\left(x \right)}}
             6         
/       2/x\\     12/x\
|1 - tan |-|| *cos  |-|
\        \2//       \2/
(1tan2(x2))6cos12(x2)\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{6} \cos^{12}{\left(\frac{x}{2} \right)}
                                                                  //                                 /           x             \\     //                                 /           x             \\                                                                                                                                                                                           //                                 /           x             \\
     //                            /           x           \\     ||           1              for And|im(x) = 0, - mod 2*pi = 0||     ||           1              for And|im(x) = 0, - mod 2*pi = 0||      //                           /           x           \\      //                             /           x           \\      //                            /           x           \\ ||           1              for And|im(x) = 0, - mod 2*pi = 0||
     ||         0           for And|im(x) = 0, - mod pi = 0||     ||                                 \           2             /|     ||                                 \           2             /|      ||        0           for And|im(x) = 0, - mod pi = 0||      ||         0            for And|im(x) = 0, - mod pi = 0||      ||         0           for And|im(x) = 0, - mod pi = 0|| ||                                 \           2             /|
     ||                            \           2           /|     ||                                                            |     ||                                                            |      ||                           \           2           /|      ||                             \           2           /|      ||                            \           2           /| ||                                                            |
- 68*|<                                                     | - 8*|<              6                                             | + 9*|<              8                                             | + 30*|<                                                    | + 39*|<                                                      | - 64*|<                                                     |*|<              6                                             |
     ||      6/x\    12/x\                                  |     ||/        2/x\\     12/x\                                    |     ||/        2/x\\     16/x\                                    |      ||      4/x\    8/x\                                  |      ||       8/x\    16/x\                                  |      ||      6/x\    12/x\                                  | ||/        2/x\\     12/x\                                    |
     ||64*cot |-|*sin  |-|             otherwise            |     |||-1 + cot |-|| *sin  |-|              otherwise             |     |||-1 + cot |-|| *sin  |-|              otherwise             |      ||16*cot |-|*sin |-|             otherwise            |      ||256*cot |-|*sin  |-|             otherwise            |      ||64*cot |-|*sin  |-|             otherwise            | |||-1 + cot |-|| *sin  |-|              otherwise             |
     \\       \4/      \4/                                  /     ||\         \4//       \4/                                    |     ||\         \4//       \4/                                    |      \\       \4/     \4/                                  /      \\        \4/      \4/                                  /      \\       \4/      \4/                                  / ||\         \4//       \4/                                    |
                                                                  \\                                                            /     \\                                                            /                                                                                                                                                                                           \\                                                            /
(30({0forim(x)=0x2modπ=016sin8(x4)cot4(x4)otherwise))(64({0forim(x)=0x2modπ=064sin12(x4)cot6(x4)otherwise)({1forim(x)=0x2mod2π=0(cot2(x4)1)6sin12(x4)otherwise))(68({0forim(x)=0x2modπ=064sin12(x4)cot6(x4)otherwise))+(39({0forim(x)=0x2modπ=0256sin16(x4)cot8(x4)otherwise))(8({1forim(x)=0x2mod2π=0(cot2(x4)1)6sin12(x4)otherwise))+(9({1forim(x)=0x2mod2π=0(cot2(x4)1)8sin16(x4)otherwise))\left(30 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod \pi = 0 \\16 \sin^{8}{\left(\frac{x}{4} \right)} \cot^{4}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right)\right) - \left(64 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod \pi = 0 \\64 \sin^{12}{\left(\frac{x}{4} \right)} \cot^{6}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{6} \sin^{12}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right)\right) - \left(68 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod \pi = 0 \\64 \sin^{12}{\left(\frac{x}{4} \right)} \cot^{6}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(39 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod \pi = 0 \\256 \sin^{16}{\left(\frac{x}{4} \right)} \cot^{8}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right)\right) - \left(8 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{6} \sin^{12}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(9 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{8} \sin^{16}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right)\right)
   6   
cos (x)
cos6(x)\cos^{6}{\left(x \right)}
       68           8         9           30             39                 64         
- ------------ - ------- + ------- + ------------ + ------------ - --------------------
     6/x   pi\      6/x\      8/x\      4/x   pi\      8/x   pi\      6/x\    6/x   pi\
  sec |- - --|   sec |-|   sec |-|   sec |- - --|   sec |- - --|   sec |-|*sec |- - --|
      \2   2 /       \2/       \2/       \2   2 /       \2   2 /       \2/     \2   2 /
30sec4(x2π2)68sec6(x2π2)+39sec8(x2π2)8sec6(x2)64sec6(x2)sec6(x2π2)+9sec8(x2)\frac{30}{\sec^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{68}{\sec^{6}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{39}{\sec^{8}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{8}{\sec^{6}{\left(\frac{x}{2} \right)}} - \frac{64}{\sec^{6}{\left(\frac{x}{2} \right)} \sec^{6}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{9}{\sec^{8}{\left(\frac{x}{2} \right)}}
                                                                                                                                                                                                         //          1             for And(im(x) = 0, x mod 2*pi = 0)\
  //   0     for And(im(x) = 0, x mod pi = 0)\   //   0     for And(im(x) = 0, x mod pi = 0)\ //   1     for And(im(x) = 0, x mod 2*pi = 0)\   //        0           for And(im(x) = 0, x mod pi = 0)\   ||                                                          |
  ||                                         |   ||                                         | ||                                           |   ||                                                    |   ||             4                                            |
- |<   6                                     | - |<   2                                     |*|<   2                                       | + |<      4/x\    4/x\                                  | + |
({0forim(x)=0xmodπ=016sin4(x2)cos4(x2)otherwise)(({0forim(x)=0xmodπ=0sin2(x)otherwise)({1forim(x)=0xmod2π=0cos2(x)otherwise))({0forim(x)=0xmodπ=0sin6(x)otherwise)+({1forim(x)=0xmod2π=0(1tan2(x2))4cos8(x2)otherwise)\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\16 \sin^{4}{\left(\frac{x}{2} \right)} \cos^{4}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{4} \cos^{8}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right)
        6/x\        6/x\        8/x\         4/x\         8/x\         6/x\    6/x\
- 68*sin |-| - 8*cos |-| + 9*cos |-| + 30*sin |-| + 39*sin |-| - 64*cos |-|*sin |-|
         \2/         \2/         \2/          \2/          \2/          \2/     \2/
39sin8(x2)64sin6(x2)cos6(x2)68sin6(x2)+30sin4(x2)+9cos8(x2)8cos6(x2)39 \sin^{8}{\left(\frac{x}{2} \right)} - 64 \sin^{6}{\left(\frac{x}{2} \right)} \cos^{6}{\left(\frac{x}{2} \right)} - 68 \sin^{6}{\left(\frac{x}{2} \right)} + 30 \sin^{4}{\left(\frac{x}{2} \right)} + 9 \cos^{8}{\left(\frac{x}{2} \right)} - 8 \cos^{6}{\left(\frac{x}{2} \right)}
                                                                                                                                               //          0            for And(im(x) = 0, x mod pi = 0)\   //                 1                    for And(im(x) = 0, x mod 2*pi = 0)\
                                                                                                                                               ||                                                       |   ||                                                                        |
  //   0     for And(im(x) = 0, x mod pi = 0)\   //   0     for And(im(x) = 0, x mod pi = 0)\ //   1     for And(im(x) = 0, x mod 2*pi = 0)\   ||        16/x\    8/x\                                  |   ||                  4                                                     |
  ||                                         |   ||                                         | ||                                           |   ||4096*cos  |-|*tan |-|                                  |   ||    /        1   \     16/x\    8/x\                                    |
- |<   6                                     | - |<   2                                     |*|<   2                                       | + |<          \4/     \4/                                  | + |<256*|-1 + -------| *cos  |-|*tan |-|              otherwise             |
  ||sin (x)             otherwise            |   ||sin (x)             otherwise            | ||cos (x)              otherwise             |   ||---------------------             otherwise            |   ||    |        2/x\|       \4/     \4/                                    |
  \\                                         /   \\                                         / \\                                           /   ||          4/x\                                         |   ||    |     tan |-||                                                      |
                                                                                                                                               ||       tan |-|                                         |   ||    \         \2//                                                      |
                                                                                                                                               \\           \2/                                         /   \\                                                                        /
({0forim(x)=0xmodπ=04096cos16(x4)tan8(x4)tan4(x2)otherwise)(({0forim(x)=0xmodπ=0sin2(x)otherwise)({1forim(x)=0xmod2π=0cos2(x)otherwise))({0forim(x)=0xmodπ=0sin6(x)otherwise)+({1forim(x)=0xmod2π=0256(1+1tan2(x2))4cos16(x4)tan8(x4)otherwise)\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{4096 \cos^{16}{\left(\frac{x}{4} \right)} \tan^{8}{\left(\frac{x}{4} \right)}}{\tan^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\256 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \cos^{16}{\left(\frac{x}{4} \right)} \tan^{8}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right)
     68          8              9            30        39              64         
- ------- - ------------ + ------------ + ------- + ------- - --------------------
     6/x\      6/pi   x\      8/pi   x\      4/x\      8/x\      6/x\    6/pi   x\
  csc |-|   csc |-- - -|   csc |-- - -|   csc |-|   csc |-|   csc |-|*csc |-- - -|
      \2/       \2    2/       \2    2/       \2/       \2/       \2/     \2    2/
8csc6(x2+π2)+9csc8(x2+π2)+30csc4(x2)68csc6(x2)64csc6(x2)csc6(x2+π2)+39csc8(x2)- \frac{8}{\csc^{6}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{9}{\csc^{8}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{30}{\csc^{4}{\left(\frac{x}{2} \right)}} - \frac{68}{\csc^{6}{\left(\frac{x}{2} \right)}} - \frac{64}{\csc^{6}{\left(\frac{x}{2} \right)} \csc^{6}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{39}{\csc^{8}{\left(\frac{x}{2} \right)}}
                                                                                                                                                                                                                        //                1                  for And(im(x) = 0, x mod 2*pi = 0)\
                                                                                                                                                                                                                        ||                                                                     |
                                                                                                                                                                                                                        ||                   4                                                 |
  //     0        for And(im(x) = 0, x mod pi = 0)\   //     0        for And(im(x) = 0, x mod pi = 0)\ //   1     for And(im(x) = 0, x mod 2*pi = 0)\   //           0             for And(im(x) = 0, x mod pi = 0)\   ||/          2/x\   \                                                  |
  ||                                              |   ||                                              | ||                                           |   ||                                                         |   |||       cos |-|   |                                                  |
- |<   6/    pi\                                  | - |<   2/    pi\                                  |*|<   2                                       | + |<      4/x\    4/x   pi\                                  | + |<|           \2/   |     8/x   pi\                                    |
  ||cos |x - --|             otherwise            |   ||cos |x - --|             otherwise            | ||cos (x)              otherwise             |   ||16*cos |-|*cos |- - --|             otherwise            |   |||-1 + ------------| *cos |- - --|              otherwise             |
  \\    \    2 /                                  /   \\    \    2 /                                  / \\                                           /   \\       \2/     \2   2 /                                  /   |||        2/x   pi\|      \2   2 /                                    |
                                                                                                                                                                                                                        |||     cos |- - --||                                                  |
                                                                                                                                                                                                                        ||\         \2   2 //                                                  |
                                                                                                                                                                                                                        \\                                                                     /
({0forim(x)=0xmodπ=016cos4(x2)cos4(x2π2)otherwise)(({0forim(x)=0xmodπ=0cos2(xπ2)otherwise)({1forim(x)=0xmod2π=0cos2(x)otherwise))({0forim(x)=0xmodπ=0cos6(xπ2)otherwise)+({1forim(x)=0xmod2π=0(cos2(x2)cos2(x2π2)1)4cos8(x2π2)otherwise)\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\16 \cos^{4}{\left(\frac{x}{2} \right)} \cos^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\cos^{2}{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\cos^{6}{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right)^{4} \cos^{8}{\left(\frac{x}{2} - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)
                                                                                                                                                                                                                                                                                //                              0                                 for And(im(x) = 0, x mod pi = 0)\   //                                 1                                   for And(im(x) = 0, x mod 2*pi = 0)\
  //                        0                          for And(im(x) = 0, x mod pi = 0)\   //                        0                          for And(im(x) = 0, x mod pi = 0)\ //                         1                            for And(im(x) = 0, x mod 2*pi = 0)\   ||                                                                                                |   ||                                                                                                       |
  ||                                                                                   |   ||                                                                                   | ||                                                                                        |   ||           //                       /           x           \\                                  |   ||                //                       /           x           \\                                    |
  ||/      0         for And(im(x) = 0, x mod pi = 0)                                  |   ||/      0         for And(im(x) = 0, x mod pi = 0)                                  | ||/       1         for And(im(x) = 0, x mod 2*pi = 0)                                    |   ||           ||      0         for And|im(x) = 0, - mod pi = 0||                                  |   ||                ||      0         for And|im(x) = 0, - mod pi = 0||                                    |
  |||                                                                                  |   |||                                                                                  | |||                                                                                       |   ||           ||                       \           2           /|                                  |   ||                ||                       \           2           /|                                    |
  |||        6/x\                                                                      |   |||       2/x\                                                                       | |||              2                                                                        |   ||           ||                                                |                                  |   ||                ||                                                |                                    |
  |||  64*cot |-|                                                                      |   |||  4*cot |-|                                                                       | |||/        2/x\\                                                                         |   ||           ||        8/x\                                    |                                  |   ||              4 ||        8/x\                                    |                                    |
- |<|         \2/                                                                      | - |<|        \2/                                                                       |*|<||-1 + cot |-||                                                                         | + |<      4/x\ || 256*cot |-|                                    |                                  | + |
({0forim(x)=0xmodπ=016({0forim(x)=0x2modπ=0256cot8(x4)(cot2(x4)+1)8otherwise)cot4(x2)otherwise)({0forim(x)=0xmodπ=0{0forim(x)=0xmodπ=064cot6(x2)(cot2(x2)+1)6otherwiseotherwise)(({0forim(x)=0xmodπ=0{0forim(x)=0xmodπ=04cot2(x2)(cot2(x2)+1)2otherwiseotherwise)({1forim(x)=0xmod2π=0{1forim(x)=0xmod2π=0(cot2(x2)1)2(cot2(x2)+1)2otherwiseotherwise))+({1forim(x)=0xmod2π=0(cot2(x2)1)4({0forim(x)=0x2modπ=0256cot8(x4)(cot2(x4)+1)8otherwise)otherwise)\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\16 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod \pi = 0 \\\frac{256 \cot^{8}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right) \cot^{4}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{64 \cot^{6}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4} \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod \pi = 0 \\\frac{256 \cot^{8}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right)
  //      0         for And(im(x) = 0, x mod pi = 0)\   //      0         for And(im(x) = 0, x mod pi = 0)\ //       1         for And(im(x) = 0, x mod 2*pi = 0)\   //      0         for And(im(x) = 0, x mod pi = 0)\   //       1         for And(im(x) = 0, x mod 2*pi = 0)\
  ||                                                |   ||                                                | ||                                                   |   ||                                                |   ||                                                   |
  ||        6/x\                                    |   ||       2/x\                                     | ||              2                                    |   ||        4/x\                                    |   ||              4                                    |
  ||  64*cot |-|                                    |   ||  4*cot |-|                                     | ||/        2/x\\                                     |   ||  16*cot |-|                                    |   ||/        2/x\\                                     |
  ||         \2/                                    |   ||        \2/                                     | |||-1 + cot |-||                                     |   ||         \2/                                    |   |||-1 + cot |-||                                     |
- |<--------------             otherwise            | - |<--------------             otherwise            |*|<\         \2//                                     | + |<--------------             otherwise            | + |<\         \2//                                     |
  ||             6                                  |   ||             2                                  | ||---------------              otherwise             |   ||             4                                  |   ||---------------              otherwise             |
  ||/       2/x\\                                   |   ||/       2/x\\                                   | ||              2                                    |   ||/       2/x\\                                   |   ||              4                                    |
  |||1 + cot |-||                                   |   |||1 + cot |-||                                   | || /       2/x\\                                     |   |||1 + cot |-||                                   |   || /       2/x\\                                     |
  ||\        \2//                                   |   ||\        \2//                                   | || |1 + cot |-||                                     |   ||\        \2//                                   |   || |1 + cot |-||                                     |
  \\                                                /   \\                                                / \\ \        \2//                                     /   \\                                                /   \\ \        \2//                                     /
({0forim(x)=0xmodπ=064cot6(x2)(cot2(x2)+1)6otherwise)+({0forim(x)=0xmodπ=016cot4(x2)(cot2(x2)+1)4otherwise)(({0forim(x)=0xmodπ=04cot2(x2)(cot2(x2)+1)2otherwise)({1forim(x)=0xmod2π=0(cot2(x2)1)2(cot2(x2)+1)2otherwise))+({1forim(x)=0xmod2π=0(cot2(x2)1)4(cot2(x2)+1)4otherwise)\left(- \begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{64 \cot^{6}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{16 \cot^{4}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right) - \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)
/       1         for And(im(x) = 0, x mod 2*pi = 0)
|                                                   
|              6                                    
|/        2/x\\                                     
||-1 + cot |-||                                     
<\         \2//                                     
|---------------              otherwise             
|              6                                    
| /       2/x\\                                     
| |1 + cot |-||                                     
\ \        \2//                                     
{1forim(x)=0xmod2π=0(cot2(x2)1)6(cot2(x2)+1)6otherwise\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{6}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}
             4                                                    2        
/       2/x\\            6/x\             4/x\       /       2/x\\     2/x\
|1 - tan |-||      64*tan |-|       16*tan |-|     4*|1 - tan |-|| *tan |-|
\        \2//             \2/              \2/       \        \2//      \2/
-------------- - -------------- + -------------- - ------------------------
             4                6                4                     4     
/       2/x\\    /       2/x\\    /       2/x\\         /       2/x\\      
|1 + tan |-||    |1 + tan |-||    |1 + tan |-||         |1 + tan |-||      
\        \2//    \        \2//    \        \2//         \        \2//      
(1tan2(x2))4(tan2(x2)+1)44(1tan2(x2))2tan2(x2)(tan2(x2)+1)4+16tan4(x2)(tan2(x2)+1)464tan6(x2)(tan2(x2)+1)6\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{4}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} - \frac{4 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2} \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} + \frac{16 \tan^{4}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} - \frac{64 \tan^{6}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}}
                                                                                                                                                                                                                                                          //                                 0                                    for And(im(x) = 0, x mod pi = 0)\   //                                    1                                      for And(im(x) = 0, x mod 2*pi = 0)\
  //                    0                       for And(im(x) = 0, x mod pi = 0)\   //                    0                       for And(im(x) = 0, x mod pi = 0)\ //                     1                        for And(im(x) = 0, x mod 2*pi = 0)\   ||                                                                                                      |   ||                                                                                                             |
  ||                                                                            |   ||                                                                            | ||                                                                                |   ||           //                             /           x           \\                                  |   ||                //                             /           x           \\                                    |
  ||/   0     for And(im(x) = 0, x mod pi = 0)                                  |   ||/   0     for And(im(x) = 0, x mod pi = 0)                                  | ||/   1     for And(im(x) = 0, x mod 2*pi = 0)                                    |   ||           ||         0            for And|im(x) = 0, - mod pi = 0||                                  |   ||              4 ||         0            for And|im(x) = 0, - mod pi = 0||                                    |
- |<|                                                                           | - |<|                                                                           |*|<|                                                                               | + |<      4/x\ ||                             \           2           /|                                  | + |
({0forim(x)=0xmodπ=016({0forim(x)=0x2modπ=0256sin16(x4)cot8(x4)otherwise)cot4(x2)otherwise)(({0forim(x)=0xmodπ=0{0forim(x)=0xmodπ=0sin2(x)otherwiseotherwise)({1forim(x)=0xmod2π=0{1forim(x)=0xmod2π=0cos2(x)otherwiseotherwise))({0forim(x)=0xmodπ=0{0forim(x)=0xmodπ=0sin6(x)otherwiseotherwise)+({1forim(x)=0xmod2π=0(cot2(x2)1)4({0forim(x)=0x2modπ=0256sin16(x4)cot8(x4)otherwise)otherwise)\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\16 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod \pi = 0 \\256 \sin^{16}{\left(\frac{x}{4} \right)} \cot^{8}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right) \cot^{4}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4} \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod \pi = 0 \\256 \sin^{16}{\left(\frac{x}{4} \right)} \cot^{8}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right)
                                  6                  8                                                       6        
           6/x\      /       2/x\\      /       2/x\\            4/x\              8/x\         /       2/x\\     6/x\
   4352*tan |-|    8*|1 - tan |-||    9*|1 - tan |-||     480*tan |-|      9984*tan |-|    4096*|1 - tan |-|| *tan |-|
            \4/      \        \4//      \        \4//             \4/               \4/         \        \4//      \4/
- -------------- - ---------------- + ---------------- + -------------- + -------------- - ---------------------------
               6                 6                  8                 4                8                      12      
  /       2/x\\     /       2/x\\      /       2/x\\     /       2/x\\    /       2/x\\          /       2/x\\        
  |1 + tan |-||     |1 + tan |-||      |1 + tan |-||     |1 + tan |-||    |1 + tan |-||          |1 + tan |-||        
  \        \4//     \        \4//      \        \4//     \        \4//    \        \4//          \        \4//        
9(1tan2(x4))8(tan2(x4)+1)88(1tan2(x4))6(tan2(x4)+1)64096(1tan2(x4))6tan6(x4)(tan2(x4)+1)12+480tan4(x4)(tan2(x4)+1)44352tan6(x4)(tan2(x4)+1)6+9984tan8(x4)(tan2(x4)+1)8\frac{9 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{8}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} - \frac{8 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{6}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}} - \frac{4096 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{6} \tan^{6}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{12}} + \frac{480 \tan^{4}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4}} - \frac{4352 \tan^{6}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}} + \frac{9984 \tan^{8}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}}
   6/    pi\
sin |x + --|
    \    2 /
sin6(x+π2)\sin^{6}{\left(x + \frac{\pi}{2} \right)}
Комбинаторика [src]
   4         4         6         2       2   
cos (x) + sin (x) - sin (x) - cos (x)*sin (x)
sin6(x)+sin4(x)sin2(x)cos2(x)+cos4(x)- \sin^{6}{\left (x \right )} + \sin^{4}{\left (x \right )} - \sin^{2}{\left (x \right )} \cos^{2}{\left (x \right )} + \cos^{4}{\left (x \right )}
Раскрыть выражение [src]
   4         4         6         2       2   
cos (x) + sin (x) - sin (x) - cos (x)*sin (x)
sin6(x)+sin4(x)sin2(x)cos2(x)+cos4(x)- \sin^{6}{\left (x \right )} + \sin^{4}{\left (x \right )} - \sin^{2}{\left (x \right )} \cos^{2}{\left (x \right )} + \cos^{4}{\left (x \right )}