sin(a)^6+cos(a)^6+3*sin(a)^2*cos(a)^2еслиa=3/2 (упростите выражение)

Вы ввели

   6         6           2       2   
sin (a) + cos (a) + 3*sin (a)*cos (a)

$$\sin^{6}{\left(a \right)} + 3 \sin^{2}{\left(a \right)} \cos^{2}{\left(a \right)} + \cos^{6}{\left(a \right)}$$

Подстановка условия

sin(a)^6 + cos(a)^6 + 3*sin(a)^2*cos(a)^2 при a = 3/2

подставляем

   6         6           2       2   
sin (a) + cos (a) + 3*sin (a)*cos (a)

$$\sin^{6}{\left(a \right)} + 3 \sin^{2}{\left(a \right)} \cos^{2}{\left(a \right)} + \cos^{6}{\left(a \right)}$$

3      6         6      3*cos(4*a)
- + cos (a) + sin (a) - ----------
8                           8     

$$\sin^{6}{\left(a \right)} + \cos^{6}{\left(a \right)} - \frac{3 \cos{\left(4 a \right)}}{8} + \frac{3}{8}$$

переменные

a = 3/2

$$a = \frac{3}{2}$$

3      6             6          3*cos(4*(3/2))
- + cos ((3/2)) + sin ((3/2)) - --------------
8                                     8       

$$\sin^{6}{\left((3/2) \right)} + \cos^{6}{\left((3/2) \right)} - \frac{3 \cos{\left(4 (3/2) \right)}}{8} + \frac{3}{8}$$

3      6           6        3*cos(4*3/2)
- + cos (3/2) + sin (3/2) - ------------
8                                8      

$$- \frac{3 \cos{\left(4 \cdot \frac{3}{2} \right)}}{8} + \cos^{6}{\left(\frac{3}{2} \right)} + \frac{3}{8} + \sin^{6}{\left(\frac{3}{2} \right)}$$

3      6           6        3*cos(6)
- + cos (3/2) + sin (3/2) - --------
8                              8    

$$- \frac{3 \cos{\left(6 \right)}}{8} + \cos^{6}{\left(\frac{3}{2} \right)} + \frac{3}{8} + \sin^{6}{\left(\frac{3}{2} \right)}$$

Степени

                                                      2                  
                                        / I*a    -I*a\                  2
              6                   6     |e      e    |  /   -I*a    I*a\ 
/ I*a    -I*a\    /   -I*a    I*a\    3*|---- + -----| *\- e     + e   / 
|e      e    |    \- e     + e   /      \ 2       2  /                   
|---- + -----|  - ----------------- - -----------------------------------
\ 2       2  /            64                           4                 

$$\left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)^{6} - \frac{3 \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)^{2} \left(e^{i a} - e^{- i a}\right)^{2}}{4} - \frac{\left(e^{i a} - e^{- i a}\right)^{6}}{64}$$

Численный ответ

cos(a)^6 + sin(a)^6 + 3.0*cos(a)^2*sin(a)^2

Рациональный знаменатель

   6         6           2       2   
cos (a) + sin (a) + 3*cos (a)*sin (a)

$$\sin^{6}{\left (a \right )} + 3 \sin^{2}{\left (a \right )} \cos^{2}{\left (a \right )} + \cos^{6}{\left (a \right )}$$

Объединение рациональных выражений

   6         6           2       2   
cos (a) + sin (a) + 3*cos (a)*sin (a)

$$\sin^{6}{\left (a \right )} + 3 \sin^{2}{\left (a \right )} \cos^{2}{\left (a \right )} + \cos^{6}{\left (a \right )}$$

Общее упрощение

3      6         6      3*cos(4*a)
- + cos (a) + sin (a) - ----------
8                           8     

$$\sin^{6}{\left(a \right)} + \cos^{6}{\left(a \right)} - \frac{3 \cos{\left(4 a \right)}}{8} + \frac{3}{8}$$

Собрать выражение

   6         6           2       2   
cos (a) + sin (a) + 3*sin (a)*cos (a)

$$\sin^{6}{\left (a \right )} + 3 \sin^{2}{\left (a \right )} \cos^{2}{\left (a \right )} + \cos^{6}{\left (a \right )}$$

1

$$1$$

Общий знаменатель

   6         6           2       2   
cos (a) + sin (a) + 3*cos (a)*sin (a)

$$\sin^{6}{\left (a \right )} + 3 \sin^{2}{\left (a \right )} \cos^{2}{\left (a \right )} + \cos^{6}{\left (a \right )}$$

Тригонометрическая часть

                 6                                     
    /       2/a\\            6/a\                      
    |1 - tan |-||      64*tan |-|       /       2     \
3   \        \2//             \2/     3*\1 - tan (2*a)/
- + -------------- + -------------- - -----------------
8                6                6     /       2     \
    /       2/a\\    /       2/a\\    8*\1 + tan (2*a)/
    |1 + tan |-||    |1 + tan |-||                     
    \        \2//    \        \2//                     

$$\frac{\left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{6}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{6}} - \frac{3 \cdot \left(1 - \tan^{2}{\left(2 a \right)}\right)}{8 \left(\tan^{2}{\left(2 a \right)} + 1\right)} + \frac{3}{8} + \frac{64 \tan^{6}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{6}}$$

                  6                                               2
/         2/a\   \                              /         2/a\   \ 
|      sec |-|   |                              |      sec |-|   | 
|          \2/   |                              |          \2/   | 
|1 - ------------|                           12*|1 - ------------| 
|       2/a   pi\|                              |       2/a   pi\| 
|    sec |- - --||                              |    sec |- - --|| 
\        \2   2 //             64               \        \2   2 // 
------------------- + -------------------- + ----------------------
         12/a\           6/a\    6/a   pi\       6/a\    2/a   pi\ 
      sec  |-|        sec |-|*sec |- - --|    sec |-|*sec |- - --| 
           \2/            \2/     \2   2 /        \2/     \2   2 / 

$$\frac{\left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{6}}{\sec^{12}{\left(\frac{a}{2} \right)}} + \frac{12 \left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}}{\sec^{6}{\left(\frac{a}{2} \right)} \sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + \frac{64}{\sec^{6}{\left(\frac{a}{2} \right)} \sec^{6}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}$$

                                                                                                                                                         //                1                   for And(im(a) = 0, a mod 2*pi = 0)\
                                                                                                                                                         ||                                                                      |
                                                                                                                                                         ||                   6                                                  |
  //     0        for And(im(a) = 0, a mod pi = 0)\ //   1     for And(im(a) = 0, a mod 2*pi = 0)\   //     0        for And(im(a) = 0, a mod pi = 0)\   ||/          2/a\   \                                                   |
  ||                                              | ||                                           |   ||                                              |   |||       cos |-|   |                                                   |
3*|<   2/    pi\                                  |*|<   2                                       | + |<   6/    pi\                                  | + |<|           \2/   |     12/a   pi\                                    |
  ||cos |a - --|             otherwise            | ||cos (a)              otherwise             |   ||cos |a - --|             otherwise            |   |||-1 + ------------| *cos  |- - --|              otherwise             |
  \\    \    2 /                                  / \\                                           /   \\    \    2 /                                  /   |||        2/a   pi\|       \2   2 /                                    |
                                                                                                                                                         |||     cos |- - --||                                                   |
                                                                                                                                                         ||\         \2   2 //                                                   |
                                                                                                                                                         \\                                                                      /

$$\left(3 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\cos^{2}{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\cos^{6}{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1\right)^{6} \cos^{12}{\left(\frac{a}{2} - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$

                                                                                                                                                                                                                                                                                //                                 1                                    for And(im(a) = 0, a mod 2*pi = 0)\
  //                        0                          for And(im(a) = 0, a mod pi = 0)\ //                         1                            for And(im(a) = 0, a mod 2*pi = 0)\   //                        0                          for And(im(a) = 0, a mod pi = 0)\   ||                                                                                                        |
  ||                                                                                   | ||                                                                                        |   ||                                                                                   |   ||                //                        /           a           \\                                    |
  ||/      0         for And(im(a) = 0, a mod pi = 0)                                  | ||/       1         for And(im(a) = 0, a mod 2*pi = 0)                                    |   ||/      0         for And(im(a) = 0, a mod pi = 0)                                  |   ||                ||       0         for And|im(a) = 0, - mod pi = 0||                                    |
  |||                                                                                  | |||                                                                                       |   |||                                                                                  |   ||                ||                        \           2           /|                                    |
  |||       2/a\                                                                       | |||              2                                                                        |   |||        6/a\                                                                      |   ||                ||                                                 |                                    |
  |||  4*cot |-|                                                                       | |||/        2/a\\                                                                         |   |||  64*cot |-|                                                                      |   ||              6 ||         12/a\                                   |                                    |
3*|<|        \2/                                                                       |*|<||-1 + cot |-||                                                                         | + |<|         \2/                                                                      | + |

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{64 \cot^{6}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(3 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{6} \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \frac{a}{2} \bmod \pi = 0 \\\frac{4096 \cot^{12}{\left(\frac{a}{4} \right)}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{12}} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right)$$

   1           1                  3          
------- + ------------ + --------------------
   6         6/pi    \      2       2/pi    \
csc (a)   csc |-- - a|   csc (a)*csc |-- - a|
              \2     /               \2     /

$$\frac{1}{\csc^{6}{\left(- a + \frac{\pi}{2} \right)}} + \frac{3}{\csc^{2}{\left(a \right)} \csc^{2}{\left(- a + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{6}{\left(a \right)}}$$

                                                                                 2                     
                                                                  /         4/a\\                      
                                                                  |    4*sin |-||                      
               6                                                  |          \2/|     4/a\    8/pi   a\
/         4/a\\                          12/a\    12/pi   a\   48*|1 - ---------| *sin |-|*sin |-- + -|
|    4*sin |-||                  4096*sin  |-|*sin  |-- + -|      |        2    |      \2/     \2    2/
|          \2/|     12/pi   a\             \2/      \2    2/      \     sin (a) /                      
|1 - ---------| *sin  |-- + -| + --------------------------- + ----------------------------------------
|        2    |       \2    2/                6                                   2                    
\     sin (a) /                            sin (a)                             sin (a)                 

$$\left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{6} \sin^{12}{\left(\frac{a}{2} + \frac{\pi}{2} \right)} + \frac{48 \left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{2} \sin^{4}{\left(\frac{a}{2} \right)} \sin^{8}{\left(\frac{a}{2} + \frac{\pi}{2} \right)}}{\sin^{2}{\left(a \right)}} + \frac{4096 \sin^{12}{\left(\frac{a}{2} \right)} \sin^{12}{\left(\frac{a}{2} + \frac{\pi}{2} \right)}}{\sin^{6}{\left(a \right)}}$$

      //            1               for And(im(a) = 0, 2*a mod pi = 0)\                                                                                                                 
      ||                                                              |                                                                                                                 
    3*|<   2      /        2     \                                    |                                                  //           1              for And(im(a) = 0, a mod 2*pi = 0)\
      ||sin (2*a)*\-1 + cot (2*a)/              otherwise             |   //   0     for And(im(a) = 0, a mod pi = 0)\   ||                                                            |
3     \\                                                              /   ||                                         |   ||              6                                             |
- - ------------------------------------------------------------------- + |<   6                                     | + |

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin^{6}{\left(a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{6} \sin^{12}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\frac{3 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\left(\cot^{2}{\left(2 a \right)} - 1\right) \sin^{2}{\left(2 a \right)} & \text{otherwise} \end{cases}\right)}{8}\right) + \frac{3}{8}$$

                                                                                                                                                                    //              1                for And(im(a) = 0, a mod 2*pi = 0)\
  //      0         for And(im(a) = 0, a mod pi = 0)\ //      1         for And(im(a) = 0, a mod 2*pi = 0)\   //      0         for And(im(a) = 0, a mod pi = 0)\   ||                                                                 |
  ||                                                | ||                                                  |   ||                                                |   ||                   6                                             |
  ||       2/a\                                     | ||             2                                    |   ||        6/a\                                    |   ||     /        1   \     12/a\                                    |
  ||  4*tan |-|                                     | ||/       2/a\\                                     |   ||  64*tan |-|                                    |   ||4096*|-1 + -------| *tan  |-|                                    |
  ||        \2/                                     | |||1 - tan |-||                                     |   ||         \2/                                    |   ||     |        2/a\|       \4/                                    |
3*|<--------------             otherwise            |*|<\        \2//                                     | + |<--------------             otherwise            | + |<     |     tan |-||                                              |
  ||             2                                  | ||--------------              otherwise             |   ||             6                                  |   ||     \         \2//                                              |
  ||/       2/a\\                                   | ||             2                                    |   ||/       2/a\\                                   |   ||-----------------------------              otherwise             |
  |||1 + tan |-||                                   | ||/       2/a\\                                     |   |||1 + tan |-||                                   |   ||                    12                                           |
  ||\        \2//                                   | |||1 + tan |-||                                     |   ||\        \2//                                   |   ||       /       2/a\\                                             |
  \\                                                / \\\        \2//                                     /   \\                                                /   ||       |1 + tan |-||                                             |
                                                                                                                                                                    \\       \        \4//                                             /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{64 \tan^{6}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right) + \left(3 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{\left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{4096 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{6} \tan^{12}{\left(\frac{a}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{12}} & \text{otherwise} \end{cases}\right)$$

      //      1         for And(im(a) = 0, 2*a mod pi = 0)\                                                                                                               
      ||                                                  |                                                                                                               
      ||        2                                         |   //      0         for And(im(a) = 0, a mod pi = 0)\   //       1         for And(im(a) = 0, a mod 2*pi = 0)\
    3*|<-1 + cot (2*a)                                    |   ||                                                |   ||                                                   |
      ||--------------              otherwise             |   ||        6/a\                                    |   ||              6                                    |
      ||       2                                          |   ||  64*cot |-|                                    |   ||/        2/a\\                                     |
3     \\1 + cot (2*a)                                     /   ||         \2/                                    |   |||-1 + cot |-||                                     |
- - ------------------------------------------------------- + |<--------------             otherwise            | + |<\         \2//                                     |
8                              8                              ||             6                                  |   ||---------------              otherwise             |
                                                              ||/       2/a\\                                   |   ||              6                                    |
                                                              |||1 + cot |-||                                   |   || /       2/a\\                                     |
                                                              ||\        \2//                                   |   || |1 + cot |-||                                     |
                                                              \\                                                /   \\ \        \2//                                     /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{64 \cot^{6}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{6}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right) - \left(\frac{3 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{\cot^{2}{\left(2 a \right)} - 1}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right)}{8}\right) + \frac{3}{8}$$

                  6                                                           2                     
/       2/a   pi\\                                          /       2/a   pi\\                      
|    cos |- - --||                                          |    cos |- - --||                      
|        \2   2 /|     12/a\         6/a\    6/a   pi\      |        \2   2 /|     6/a\    2/a   pi\
|1 - ------------| *cos  |-| + 64*cos |-|*cos |- - --| + 12*|1 - ------------| *cos |-|*cos |- - --|
|         2/a\   |       \2/          \2/     \2   2 /      |         2/a\   |      \2/     \2   2 /
|      cos |-|   |                                          |      cos |-|   |                      
\          \2/   /                                          \          \2/   /                      

$$\left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)^{6} \cos^{12}{\left(\frac{a}{2} \right)} + 12 \left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \cos^{6}{\left(\frac{a}{2} \right)} \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)} + 64 \cos^{6}{\left(\frac{a}{2} \right)} \cos^{6}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}$$

   6         6/    pi\        2       2/    pi\
sin (a) + sin |a + --| + 3*sin (a)*sin |a + --|
              \    2 /                 \    2 /

$$\sin^{6}{\left(a \right)} + 3 \sin^{2}{\left(a \right)} \sin^{2}{\left(a + \frac{\pi}{2} \right)} + \sin^{6}{\left(a + \frac{\pi}{2} \right)}$$

                                                                                                                                                         //         1            for And(im(a) = 0, a mod 2*pi = 0)\
                                                                                                                                                         ||                                                        |
                                                                                                                                                         ||                   6                                    |
                                                                                                                                                         ||/        2/a   pi\\                                     |
  //     0        for And(im(a) = 0, a mod pi = 0)\ //   1     for And(im(a) = 0, a mod 2*pi = 0)\   //     0        for And(im(a) = 0, a mod pi = 0)\   |||     sec |- - --||                                     |
  ||                                              | ||                                           |   ||                                              |   |||         \2   2 /|                                     |
  ||     1                                        | ||   1                                       |   ||     1                                        |   |||-1 + ------------|                                     |
3*|<------------             otherwise            |*|<-------              otherwise             | + |<------------             otherwise            | + |<|          2/a\   |                                     |
  ||   2/    pi\                                  | ||   2                                       |   ||   6/    pi\                                  |   |||       sec |-|   |                                     |
  ||sec |a - --|                                  | ||sec (a)                                    |   ||sec |a - --|                                  |   ||\           \2/   /                                     |
  \\    \    2 /                                  / \\                                           /   \\    \    2 /                                  /   ||--------------------              otherwise             |
                                                                                                                                                         ||      12/a   pi\                                        |
                                                                                                                                                         ||   sec  |- - --|                                        |
                                                                                                                                                         ||        \2   2 /                                        |
                                                                                                                                                         \\                                                        /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1}{\sec^{6}{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(3 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right)^{6}}{\sec^{12}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$

             6              12                   12                           2              8        
/       2/a\\  /       2/a\\        /       2/a\\      6/a\      /       2/a\\  /       2/a\\     2/a\
|1 - tan |-|| *|1 - tan |-||     64*|1 - tan |-||  *tan |-|   12*|1 - tan |-|| *|1 - tan |-|| *tan |-|
\        \2//  \        \4//        \        \4//       \2/      \        \2//  \        \4//      \2/
------------------------------ + -------------------------- + ----------------------------------------
                    12                             12                                   8             
       /       2/a\\                  /       2/a\\                        /       2/a\\              
       |1 + tan |-||                  |1 + tan |-||                        |1 + tan |-||              
       \        \4//                  \        \4//                        \        \4//              

$$\frac{\left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{12} \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{6}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{12}} + \frac{64 \left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{12} \tan^{6}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{12}} + \frac{12 \left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{8} \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{2} \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{8}}$$

3      1           1                3       
- + ------- + ------------ - ---------------
8      6         6/pi    \        /pi      \
    csc (a)   csc |-- - a|   8*csc|-- - 4*a|
                  \2     /        \2       /

$$\frac{3}{8} + \frac{1}{\csc^{6}{\left(- a + \frac{\pi}{2} \right)}} - \frac{3}{8 \csc{\left(- 4 a + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{6}{\left(a \right)}}$$

3      6         6/    pi\   3*cos(4*a)
- + cos (a) + cos |a - --| - ----------
8                 \    2 /       8     

$$\cos^{6}{\left(a \right)} - \frac{3 \cos{\left(4 a \right)}}{8} + \cos^{6}{\left(a - \frac{\pi}{2} \right)} + \frac{3}{8}$$

   1         1             3       
------- + ------- + ---------------
   6         6         2       2   
csc (a)   sec (a)   csc (a)*sec (a)

$$\frac{1}{\sec^{6}{\left(a \right)}} + \frac{3}{\csc^{2}{\left(a \right)} \sec^{2}{\left(a \right)}} + \frac{1}{\csc^{6}{\left(a \right)}}$$

                                                                                                                                               //           1              for And(im(a) = 0, a mod 2*pi = 0)\
  //   0     for And(im(a) = 0, a mod pi = 0)\ //   1     for And(im(a) = 0, a mod 2*pi = 0)\   //   0     for And(im(a) = 0, a mod pi = 0)\   ||                                                            |
  ||                                         | ||                                           |   ||                                         |   ||              6                                             |
3*|<   2                                     |*|<   2                                       | + |<   6                                     | + |

$$\left(3 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin^{6}{\left(a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{6} \sin^{12}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right)$$

             6                                                  2                
/       2/a\\     12/a\         12/a\    6/a\      /       2/a\\     8/a\    2/a\
|1 - tan |-|| *cos  |-| + 64*cos  |-|*tan |-| + 12*|1 - tan |-|| *cos |-|*tan |-|
\        \2//       \2/           \2/     \2/      \        \2//      \2/     \2/

$$\left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{6} \cos^{12}{\left(\frac{a}{2} \right)} + 12 \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{2} \cos^{8}{\left(\frac{a}{2} \right)} \tan^{2}{\left(\frac{a}{2} \right)} + 64 \cos^{12}{\left(\frac{a}{2} \right)} \tan^{6}{\left(\frac{a}{2} \right)}$$

3      6         6          3     
- + cos (a) + sin (a) - ----------
8                       8*sec(4*a)

$$\sin^{6}{\left(a \right)} + \cos^{6}{\left(a \right)} + \frac{3}{8} - \frac{3}{8 \sec{\left(4 a \right)}}$$

                             /pi      \
                        3*sin|-- + 4*a|
3      6         6           \2       /
- + cos (a) + sin (a) - ---------------
8                              8       

$$\sin^{6}{\left(a \right)} - \frac{3 \sin{\left(4 a + \frac{\pi}{2} \right)}}{8} + \cos^{6}{\left(a \right)} + \frac{3}{8}$$

               //                                  /           a             \\              //                                  /           a             \\                             //                                 /           a             \\
               ||            1              for And|im(a) = 0, - mod 2*pi = 0||              ||            1              for And|im(a) = 0, - mod 2*pi = 0||                             ||           1              for And|im(a) = 0, - mod 2*pi = 0||
             6 ||                                  \           2             /|              ||                                  \           2             /|                   2         ||                                 \           2             /|
/       2/a\\  ||                                                             |         6/a\ ||                                                             |      /       2/a\\     2/a\ ||                                                            |
|1 - tan |-|| *|<              12                                             | + 64*tan |-|*|<              12                                             | + 12*|1 - tan |-|| *tan |-|*|<              8                                             |
\        \2//  ||/        2/a\\      24/a\                                    |          \2/ ||/        2/a\\      24/a\                                    |      \        \2//      \2/ ||/        2/a\\     16/a\                                    |
               |||-1 + cot |-||  *sin  |-|              otherwise             |              |||-1 + cot |-||  *sin  |-|              otherwise             |                             |||-1 + cot |-|| *sin  |-|              otherwise             |
               ||\         \4//        \4/                                    |              ||\         \4//        \4/                                    |                             ||\         \4//       \4/                                    |
               \\                                                             /              \\                                                             /                             \\                                                            /

$$\left(\left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{6} \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \frac{a}{2} \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{12} \sin^{24}{\left(\frac{a}{4} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(12 \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{2} \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \frac{a}{2} \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{8} \sin^{16}{\left(\frac{a}{4} \right)} & \text{otherwise} \end{cases}\right) \tan^{2}{\left(\frac{a}{2} \right)}\right) + \left(64 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \frac{a}{2} \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{12} \sin^{24}{\left(\frac{a}{4} \right)} & \text{otherwise} \end{cases}\right) \tan^{6}{\left(\frac{a}{2} \right)}\right)$$

                                                                                                                                               //            1               for And(im(a) = 0, a mod 2*pi = 0)\
                                                                                                                                               ||                                                              |
  //   0     for And(im(a) = 0, a mod pi = 0)\ //   1     for And(im(a) = 0, a mod 2*pi = 0)\   //   0     for And(im(a) = 0, a mod pi = 0)\   ||                6                                             |
  ||                                         | ||                                           |   ||                                         |   ||/         2    \                                              |
3*|<   2                                     |*|<   2                                       | + |<   6                                     | + |<|      sin (a) |     12/a\                                    |
  ||sin (a)             otherwise            | ||cos (a)              otherwise             |   ||sin (a)             otherwise            |   |||-1 + ---------| *sin  |-|              otherwise             |
  \\                                         / \\                                           /   \\                                         /   |||          4/a\|       \2/                                    |
                                                                                                                                               |||     4*sin |-||                                              |
                                                                                                                                               \\\           \2//                                              /

$$\left(3 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin^{6}{\left(a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\left(-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{6} \sin^{12}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right)$$

                                                                                                                                               //         1           for And(im(a) = 0, a mod 2*pi = 0)\
                                                                                                                                               ||                                                       |
  //   0     for And(im(a) = 0, a mod pi = 0)\ //   1     for And(im(a) = 0, a mod 2*pi = 0)\   //   0     for And(im(a) = 0, a mod pi = 0)\   ||      6       12/a\                                    |
  ||                                         | ||                                           |   ||                                         |   ||64*cos (a)*sin  |-|                                    |
3*|<   2                                     |*|<   2                                       | + |<   6                                     | + |<                \2/                                    |
  ||sin (a)             otherwise            | ||cos (a)              otherwise             |   ||sin (a)             otherwise            |   ||-------------------              otherwise             |
  \\                                         / \\                                           /   \\                                         /   ||                6                                      |
                                                                                                                                               ||   (-1 + cos(a))                                       |
                                                                                                                                               \\                                                       /

$$\left(3 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin^{6}{\left(a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{64 \sin^{12}{\left(\frac{a}{2} \right)} \cos^{6}{\left(a \right)}}{\left(\cos{\left(a \right)} - 1\right)^{6}} & \text{otherwise} \end{cases}\right)$$

                                              6       12/a\
                                        64*cos (a)*cos  |-|
3*(1 - cos(4*a))         6/a\    6/a\                   \2/
---------------- + 64*cos |-|*sin |-| + -------------------
       8                  \2/     \2/                  6   
                                           (1 + cos(a))    

$$\frac{3 \cdot \left(1 - \cos{\left(4 a \right)}\right)}{8} + 64 \sin^{6}{\left(\frac{a}{2} \right)} \cos^{6}{\left(\frac{a}{2} \right)} + \frac{64 \cos^{12}{\left(\frac{a}{2} \right)} \cos^{6}{\left(a \right)}}{\left(\cos{\left(a \right)} + 1\right)^{6}}$$

   1           1                  3          
------- + ------------ + --------------------
   6         6/    pi\      2       2/    pi\
sec (a)   sec |a - --|   sec (a)*sec |a - --|
              \    2 /               \    2 /

$$\frac{1}{\sec^{6}{\left(a - \frac{\pi}{2} \right)}} + \frac{3}{\sec^{2}{\left(a \right)} \sec^{2}{\left(a - \frac{\pi}{2} \right)}} + \frac{1}{\sec^{6}{\left(a \right)}}$$

                  6                                               2
/       2/pi   a\\                              /       2/pi   a\\ 
|    csc |-- - -||                              |    csc |-- - -|| 
|        \2    2/|                              |        \2    2/| 
|1 - ------------|                           12*|1 - ------------| 
|         2/a\   |                              |         2/a\   | 
|      csc |-|   |                              |      csc |-|   | 
\          \2/   /             64               \          \2/   / 
------------------- + -------------------- + ----------------------
      12/pi   a\         6/a\    6/pi   a\       2/a\    6/pi   a\ 
   csc  |-- - -|      csc |-|*csc |-- - -|    csc |-|*csc |-- - -| 
        \2    2/          \2/     \2    2/        \2/     \2    2/ 

$$\frac{\left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)^{6}}{\csc^{12}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + \frac{12 \left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)^{2}}{\csc^{2}{\left(\frac{a}{2} \right)} \csc^{6}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + \frac{64}{\csc^{6}{\left(\frac{a}{2} \right)} \csc^{6}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}$$

                                                                                                                                               //                  1                     for And(im(a) = 0, a mod 2*pi = 0)\
                                                                                                                                               ||                                                                          |
  //   0     for And(im(a) = 0, a mod pi = 0)\ //   1     for And(im(a) = 0, a mod 2*pi = 0)\   //   0     for And(im(a) = 0, a mod pi = 0)\   ||                   6                                                      |
  ||                                         | ||                                           |   ||                                         |   ||     /        1   \     24/a\    12/a\                                    |
3*|<   2                                     |*|<   2                                       | + |<   6                                     | + |<4096*|-1 + -------| *cos  |-|*tan  |-|              otherwise             |
  ||sin (a)             otherwise            | ||cos (a)              otherwise             |   ||sin (a)             otherwise            |   ||     |        2/a\|       \4/      \4/                                    |
  \\                                         / \\                                           /   \\                                         /   ||     |     tan |-||                                                       |
                                                                                                                                               ||     \         \2//                                                       |
                                                                                                                                               \\                                                                          /

$$\left(3 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin^{6}{\left(a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\4096 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{6} \cos^{24}{\left(\frac{a}{4} \right)} \tan^{12}{\left(\frac{a}{4} \right)} & \text{otherwise} \end{cases}\right)$$

                                  /pi      \
                             3*sin|-- + 4*a|
3      6         6/    pi\        \2       /
- + sin (a) + sin |a + --| - ---------------
8                 \    2 /          8       

$$\sin^{6}{\left(a \right)} + \sin^{6}{\left(a + \frac{\pi}{2} \right)} - \frac{3 \sin{\left(4 a + \frac{\pi}{2} \right)}}{8} + \frac{3}{8}$$

       6            6/a\    6/a\
1 - sin (a) + 64*cos |-|*sin |-|
                     \2/     \2/

$$64 \sin^{6}{\left(\frac{a}{2} \right)} \cos^{6}{\left(\frac{a}{2} \right)} - \sin^{6}{\left(a \right)} + 1$$

3      6         6             3       
- + cos (a) + sin (a) - ---------------
8                            /pi      \
                        8*csc|-- - 4*a|
                             \2       /

$$\sin^{6}{\left(a \right)} + \cos^{6}{\left(a \right)} + \frac{3}{8} - \frac{3}{8 \csc{\left(- 4 a + \frac{\pi}{2} \right)}}$$

                                                                            //                         /           a             \\                     //                        /           a             \\
                                                                            ||       1          for And|im(a) = 0, - mod 2*pi = 0||                     ||       1         for And|im(a) = 0, - mod 2*pi = 0||
                                                                            ||                         \           2             /|                     ||                        \           2             /|
                                                                            ||                                                    |                     ||                                                   |
                                                                            ||              12                                    |                   2 ||              8                                    |
                                                                            ||/        2/a\\                                      |      /       1   \  ||/        2/a\\                                     |
                                                                         64*|<|-1 + cot |-||                                      |   12*|1 - -------| *|<|-1 + cot |-||                                     |
               //                         /           a             \\      ||\         \4//                                      |      |       2/a\|  ||\         \4//                                     |
               ||       1          for And|im(a) = 0, - mod 2*pi = 0||      ||----------------              otherwise             |      |    cot |-||  ||---------------              otherwise             |
               ||                         \           2             /|      ||             12                                     |      \        \2//  ||              8                                    |
               ||                                                    |      ||/       2/a\\                                       |                     || /       2/a\\                                     |
             6 ||              12                                    |      |||1 + cot |-||                                       |                     || |1 + cot |-||                                     |
/       1   \  ||/        2/a\\                                      |      \\\        \4//                                       /                     \\ \        \4//                                     /
|1 - -------| *|<|-1 + cot |-||                                      | + ---------------------------------------------------------- + ------------------------------------------------------------------------
|       2/a\|  ||\         \4//                                      |                               6/a\                                                                2/a\                                 
|    cot |-||  ||----------------              otherwise             |                            cot |-|                                                             cot |-|                                 
\        \2//  ||             12                                     |                                \2/                                                                 \2/                                 
               ||/       2/a\\                                       |                                                                                                                                        
               |||1 + cot |-||                                       |                                                                                                                                        
               \\\        \4//                                       /                                                                                                                                        

$$\left(\left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{6} \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \frac{a}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{12}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{12}} & \text{otherwise} \end{cases}\right)\right) + \left(\frac{12 \left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \frac{a}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{8}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right)}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) + \left(\frac{64 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \frac{a}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{12}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{12}} & \text{otherwise} \end{cases}\right)}{\cot^{6}{\left(\frac{a}{2} \right)}}\right)$$

             6                                                 2                
/       2/a\\     12/a\         6/a\    6/a\      /       2/a\\     6/a\    2/a\
|1 - tan |-|| *cos  |-| + 64*cos |-|*sin |-| + 12*|1 - tan |-|| *cos |-|*sin |-|
\        \2//       \2/          \2/     \2/      \        \2//      \2/     \2/

$$\left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{6} \cos^{12}{\left(\frac{a}{2} \right)} + 12 \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{2} \sin^{2}{\left(\frac{a}{2} \right)} \cos^{6}{\left(\frac{a}{2} \right)} + 64 \sin^{6}{\left(\frac{a}{2} \right)} \cos^{6}{\left(\frac{a}{2} \right)}$$

  //      0         for And(im(a) = 0, a mod pi = 0)\ //       1         for And(im(a) = 0, a mod 2*pi = 0)\   //      0         for And(im(a) = 0, a mod pi = 0)\   //       1         for And(im(a) = 0, a mod 2*pi = 0)\
  ||                                                | ||                                                   |   ||                                                |   ||                                                   |
  ||       2/a\                                     | ||              2                                    |   ||        6/a\                                    |   ||              6                                    |
  ||  4*cot |-|                                     | ||/        2/a\\                                     |   ||  64*cot |-|                                    |   ||/        2/a\\                                     |
  ||        \2/                                     | |||-1 + cot |-||                                     |   ||         \2/                                    |   |||-1 + cot |-||                                     |
3*|<--------------             otherwise            |*|<\         \2//                                     | + |<--------------             otherwise            | + |<\         \2//                                     |
  ||             2                                  | ||---------------              otherwise             |   ||             6                                  |   ||---------------              otherwise             |
  ||/       2/a\\                                   | ||              2                                    |   ||/       2/a\\                                   |   ||              6                                    |
  |||1 + cot |-||                                   | || /       2/a\\                                     |   |||1 + cot |-||                                   |   || /       2/a\\                                     |
  ||\        \2//                                   | || |1 + cot |-||                                     |   ||\        \2//                                   |   || |1 + cot |-||                                     |
  \\                                                / \\ \        \2//                                     /   \\                                                /   \\ \        \2//                                     /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{64 \cot^{6}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right) + \left(3 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{6}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right)$$

3      1           1             3     
- + ------- + ------------ - ----------
8      6         6/    pi\   8*sec(4*a)
    sec (a)   sec |a - --|             
                  \    2 /             

$$\frac{3}{8} + \frac{1}{\sec^{6}{\left(a - \frac{\pi}{2} \right)}} - \frac{3}{8 \sec{\left(4 a \right)}} + \frac{1}{\sec^{6}{\left(a \right)}}$$

                 6                                       2      /       2     \
3   /       2/a\\     12/a\         12/a\    6/a\   3*cos (2*a)*\1 - tan (2*a)/
- + |1 - tan |-|| *cos  |-| + 64*cos  |-|*tan |-| - ---------------------------
8   \        \2//       \2/           \2/     \2/                8             

$$\left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{6} \cos^{12}{\left(\frac{a}{2} \right)} - \frac{3 \cdot \left(1 - \tan^{2}{\left(2 a \right)}\right) \cos^{2}{\left(2 a \right)}}{8} + 64 \cos^{12}{\left(\frac{a}{2} \right)} \tan^{6}{\left(\frac{a}{2} \right)} + \frac{3}{8}$$

                                                                                                                                               //           1             for And(im(a) = 0, a mod 2*pi = 0)\
  //   0     for And(im(a) = 0, a mod pi = 0)\ //   1     for And(im(a) = 0, a mod 2*pi = 0)\   //   0     for And(im(a) = 0, a mod pi = 0)\   ||                                                           |
  ||                                         | ||                                           |   ||                                         |   ||             6                                             |
3*|<   2                                     |*|<   2                                       | + |<   6                                     | + |

$$\left(3 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin^{6}{\left(a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{6} \cos^{12}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right)$$

3      6         6      3*cos(4*a)
- + cos (a) + sin (a) - ----------
8                           8     

$$\sin^{6}{\left(a \right)} + \cos^{6}{\left(a \right)} - \frac{3 \cos{\left(4 a \right)}}{8} + \frac{3}{8}$$

                                                                                                                                                    //         1            for And(im(a) = 0, a mod 2*pi = 0)\
                                                                                                                                                    ||                                                        |
                                                                                                                                                    ||                   6                                    |
                                                                                                                                                    ||/          2/a\   \                                     |
  //   0     for And(im(a) = 0, a mod pi = 0)\ //     1        for And(im(a) = 0, a mod 2*pi = 0)\   //   0     for And(im(a) = 0, a mod pi = 0)\   |||       csc |-|   |                                     |
  ||                                         | ||                                                |   ||                                         |   |||           \2/   |                                     |
  ||   1                                     | ||     1                                          |   ||   1                                     |   |||-1 + ------------|                                     |
3*|<-------             otherwise            |*|<------------              otherwise             | + |<-------             otherwise            | + |<|        2/pi   a\|                                     |
  ||   2                                     | ||   2/pi    \                                    |   ||   6                                     |   |||     csc |-- - -||                                     |
  ||csc (a)                                  | ||csc |-- - a|                                    |   ||csc (a)                                  |   ||\         \2    2//                                     |
  \\                                         / \\    \2     /                                    /   \\                                         /   ||--------------------              otherwise             |
                                                                                                                                                    ||         12/a\                                          |
                                                                                                                                                    ||      csc  |-|                                          |
                                                                                                                                                    ||           \2/                                          |
                                                                                                                                                    \\                                                        /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1}{\csc^{6}{\left(a \right)}} & \text{otherwise} \end{cases}\right) + \left(3 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1\right)^{6}}{\csc^{12}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)$$

   6         6/    pi\        2       2/    pi\
cos (a) + cos |a - --| + 3*cos (a)*cos |a - --|
              \    2 /                 \    2 /

$$\cos^{6}{\left(a \right)} + 3 \cos^{2}{\left(a \right)} \cos^{2}{\left(a - \frac{\pi}{2} \right)} + \cos^{6}{\left(a - \frac{\pi}{2} \right)}$$

                                                                                                                                                                                                                                                          //                                     1                                       for And(im(a) = 0, a mod 2*pi = 0)\
  //                    0                       for And(im(a) = 0, a mod pi = 0)\ //                     1                        for And(im(a) = 0, a mod 2*pi = 0)\   //                    0                       for And(im(a) = 0, a mod pi = 0)\   ||                                                                                                               |
  ||                                                                            | ||                                                                                |   ||                                                                            |   ||                //                               /           a           \\                                    |
  ||/   0     for And(im(a) = 0, a mod pi = 0)                                  | ||/   1     for And(im(a) = 0, a mod 2*pi = 0)                                    |   ||/   0     for And(im(a) = 0, a mod pi = 0)                                  |   ||              6 ||          0             for And|im(a) = 0, - mod pi = 0||                                    |
3*|<|                                                                           |*|<|                                                                               | + |<|                                                                           | + |

$$\left(3 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin^{6}{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{6} \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \frac{a}{2} \bmod \pi = 0 \\4096 \sin^{24}{\left(\frac{a}{4} \right)} \cot^{12}{\left(\frac{a}{4} \right)} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right)$$

             6              12                            12                                    2              8                 
/       2/a\\  /       2/a\\      24/a\      /       2/a\\      24/a\    6/a\      /       2/a\\  /       2/a\\     16/a\    2/a\
|1 - tan |-|| *|1 - tan |-||  *cos  |-| + 64*|1 - tan |-||  *cos  |-|*tan |-| + 12*|1 - tan |-|| *|1 - tan |-|| *cos  |-|*tan |-|
\        \2//  \        \4//        \4/      \        \4//        \4/     \2/      \        \2//  \        \4//       \4/     \2/

$$\left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{12} \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{6} \cos^{24}{\left(\frac{a}{4} \right)} + 64 \left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{12} \cos^{24}{\left(\frac{a}{4} \right)} \tan^{6}{\left(\frac{a}{2} \right)} + 12 \left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{8} \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{2} \cos^{16}{\left(\frac{a}{4} \right)} \tan^{2}{\left(\frac{a}{2} \right)}$$

             6                                    2        
/       2/a\\            6/a\        /       2/a\\     2/a\
|1 - tan |-||      64*tan |-|     12*|1 - tan |-|| *tan |-|
\        \2//             \2/        \        \2//      \2/
-------------- + -------------- + -------------------------
             6                6                      4     
/       2/a\\    /       2/a\\          /       2/a\\      
|1 + tan |-||    |1 + tan |-||          |1 + tan |-||      
\        \2//    \        \2//          \        \2//      

$$\frac{\left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{6}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{6}} + \frac{12 \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{2} \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}} + \frac{64 \tan^{6}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{6}}$$

   6         6      3*(1 - cos(4*a))
cos (a) + sin (a) + ----------------
                           8        

$$\frac{3 \cdot \left(1 - \cos{\left(4 a \right)}\right)}{8} + \sin^{6}{\left(a \right)} + \cos^{6}{\left(a \right)}$$

                                                                                                                                                    //            1               for And(im(a) = 0, a mod 2*pi = 0)\
                                                                                                                                                    ||                                                              |
  //   0     for And(im(a) = 0, a mod pi = 0)\ //     1        for And(im(a) = 0, a mod 2*pi = 0)\   //   0     for And(im(a) = 0, a mod pi = 0)\   ||                6                                             |
  ||                                         | ||                                                |   ||                                         |   ||/         2    \                                              |
3*|<   2                                     |*|<   2/    pi\                                    | + |<   6                                     | + |<|      sin (a) |     12/a\                                    |
  ||sin (a)             otherwise            | ||sin |a + --|              otherwise             |   ||sin (a)             otherwise            |   |||-1 + ---------| *sin  |-|              otherwise             |
  \\                                         / \\    \    2 /                                    /   \\                                         /   |||          4/a\|       \2/                                    |
                                                                                                                                                    |||     4*sin |-||                                              |
                                                                                                                                                    \\\           \2//                                              /

$$\left(3 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\sin^{2}{\left(a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin^{6}{\left(a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\left(-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{6} \sin^{12}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right)$$

Комбинаторика

   6         6           2       2   
cos (a) + sin (a) + 3*cos (a)*sin (a)

$$\sin^{6}{\left (a \right )} + 3 \sin^{2}{\left (a \right )} \cos^{2}{\left (a \right )} + \cos^{6}{\left (a \right )}$$

Раскрыть выражение

   6         6           2       2   
cos (a) + sin (a) + 3*cos (a)*sin (a)

$$\sin^{6}{\left (a \right )} + 3 \sin^{2}{\left (a \right )} \cos^{2}{\left (a \right )} + \cos^{6}{\left (a \right )}$$