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5*cos(x)*sin(2*x)+5*cos(2*x)*sin(x)еслиx=-1 (упростите выражение)

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Примеры

Решение

Вы ввели [src]
5*cos(x)*sin(2*x) + 5*cos(2*x)*sin(x)
$$5 \sin{\left(x \right)} \cos{\left(2 x \right)} + 5 \sin{\left(2 x \right)} \cos{\left(x \right)}$$
Подстановка условия [src]
5*cos(x)*sin(2*x) + 5*cos(2*x)*sin(x) при x = -1
подставляем
5*cos(x)*sin(2*x) + 5*cos(2*x)*sin(x)
$$5 \sin{\left(x \right)} \cos{\left(2 x \right)} + 5 \sin{\left(2 x \right)} \cos{\left(x \right)}$$
5*sin(3*x)
$$5 \sin{\left(3 x \right)}$$
переменные
x = -1
$$x = -1$$
5*sin(3*(-1))
$$5 \sin{\left(3 (-1) \right)}$$
-5*sin(3)
$$- 5 \sin{\left(3 \right)}$$
Степени [src]
      / I*x    -I*x\                            / -2*I*x    2*I*x\                 
      |e      e    | /   -2*I*x    2*I*x\       |e         e     | /   -I*x    I*x\
  5*I*|---- + -----|*\- e       + e     /   5*I*|------- + ------|*\- e     + e   /
      \ 2       2  /                            \   2        2   /                 
- --------------------------------------- - ---------------------------------------
                     2                                         2
$$- \frac{5 i \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right) \left(e^{2 i x} - e^{- 2 i x}\right)}{2} - \frac{5 i \left(e^{i x} - e^{- i x}\right) \left(\frac{e^{2 i x}}{2} + \frac{e^{- 2 i x}}{2}\right)}{2}$$
Численный ответ [src]
5.0*cos(x)*sin(2*x) + 5.0*cos(2*x)*sin(x)
Рациональный знаменатель [src]
5*cos(x)*sin(2*x) + 5*cos(2*x)*sin(x)
$$5 \sin{\left (x \right )} \cos{\left (2 x \right )} + 5 \sin{\left (2 x \right )} \cos{\left (x \right )}$$
Объединение рациональных выражений [src]
5*(cos(x)*sin(2*x) + cos(2*x)*sin(x))
$$5 \left(\sin{\left(x \right)} \cos{\left(2 x \right)} + \sin{\left(2 x \right)} \cos{\left(x \right)}\right)$$
Общее упрощение [src]
5*sin(3*x)
$$5 \sin{\left(3 x \right)}$$
Собрать выражение [src]
5*sin(3*x)
$$5 \sin{\left (3 x \right )}$$
Общий знаменатель [src]
5*cos(x)*sin(2*x) + 5*cos(2*x)*sin(x)
$$5 \sin{\left (x \right )} \cos{\left (2 x \right )} + 5 \sin{\left (2 x \right )} \cos{\left (x \right )}$$
Тригонометрическая часть [src]
         5                      5          
-------------------- + --------------------
          /      pi\               /    pi\
sec(x)*sec|2*x - --|   sec(2*x)*sec|x - --|
          \      2 /               \    2 /
$$\frac{5}{\sec{\left(2 x \right)} \sec{\left(x - \frac{\pi}{2} \right)}} + \frac{5}{\sec{\left(x \right)} \sec{\left(2 x - \frac{\pi}{2} \right)}}$$
         5                      5          
-------------------- + --------------------
          /pi      \               /pi    \
csc(x)*csc|-- - 2*x|   csc(2*x)*csc|-- - x|
          \2       /               \2     /
$$\frac{5}{\csc{\left(2 x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{5}{\csc{\left(x \right)} \csc{\left(- 2 x + \frac{\pi}{2} \right)}}$$
  //     0       for And(im(x) = 0, x mod pi = 0)\                                                   //      0        for And(im(x) = 0, 2*x mod pi = 0)\                                              
  ||                                             | //   1      for And(im(x) = 0, x mod pi = 0)\     ||                                                 | //  1     for And(im(x) = 0, x mod 2*pi = 0)\
5*|<   /    pi\                                  |*|<                                          | + 5*|<   /      pi\                                    |*|<                                          |
  ||cos|x - --|             otherwise            | \\cos(2*x)             otherwise            /     ||cos|2*x - --|              otherwise             | \\cos(x)              otherwise             /
  \\   \    2 /                                  /                                                   \\   \      2 /                                    /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\cos{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\cos{\left(2 x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right)\right)$$
  //     0       for And(im(x) = 0, x mod pi = 0)\                                                                                                         //     1       for And(im(x) = 0, x mod 2*pi = 0)\
  ||                                             | //     1       for And(im(x) = 0, x mod pi = 0)\     //     0       for And(im(x) = 0, 2*x mod pi = 0)\ ||                                               |
  ||       /x\                                   | ||                                             |     ||                                               | ||       2/x\                                    |
  ||  2*tan|-|                                   | ||       2                                     |     ||  2*tan(x)                                     | ||1 - tan |-|                                    |
5*|<       \2/                                   |*|<1 - tan (x)                                  | + 5*|<-----------              otherwise             |*|<        \2/                                    |
  ||-----------             otherwise            | ||-----------             otherwise            |     ||       2                                       | ||-----------              otherwise             |
  ||       2/x\                                  | ||       2                                     |     ||1 + tan (x)                                    | ||       2/x\                                    |
  ||1 + tan |-|                                  | \\1 + tan (x)                                  /     \\                                               / ||1 + tan |-|                                    |
  \\        \2/                                  /                                                                                                         \\        \2/                                    /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
   5           5                 5          
-------- + ---------- + --------------------
2*csc(x)   2*csc(3*x)             /pi      \
                        csc(x)*csc|-- - 2*x|
                                  \2       /
$$\frac{5}{2 \csc{\left(3 x \right)}} + \frac{5}{2 \csc{\left(x \right)}} + \frac{5}{\csc{\left(x \right)} \csc{\left(- 2 x + \frac{\pi}{2} \right)}}$$
   5    
--------
csc(3*x)
$$\frac{5}{\csc{\left(3 x \right)}}$$
5*sin(x)   5*sin(3*x)                    
-------- + ---------- + 5*cos(2*x)*sin(x)
   2           2
$$5 \sin{\left(x \right)} \cos{\left(2 x \right)} + \frac{5 \sin{\left(x \right)}}{2} + \frac{5 \sin{\left(3 x \right)}}{2}$$
     /       2   \    /x\          /       2/x\\         
  10*\1 - tan (x)/*tan|-|       10*|1 - tan |-||*tan(x)  
                      \2/          \        \2//         
--------------------------- + ---------------------------
/       2   \ /       2/x\\   /       2   \ /       2/x\\
\1 + tan (x)/*|1 + tan |-||   \1 + tan (x)/*|1 + tan |-||
              \        \2//                 \        \2//
$$\frac{10 \cdot \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \tan{\left(x \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\tan^{2}{\left(x \right)} + 1\right)} + \frac{10 \cdot \left(1 - \tan^{2}{\left(x \right)}\right) \tan{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\tan^{2}{\left(x \right)} + 1\right)}$$
     /      pi\
5*cos|3*x - --|
     \      2 /
$$5 \cos{\left(3 x - \frac{\pi}{2} \right)}$$
                                              //      1        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, 2*x mod pi = 0)\ ||                                               |
  ||                                        | ||      1                                        |     ||                                            | ||     1                                         |
5*|<  1                                     |*|<-------------             otherwise            | + 5*|<   1                                        |*|<-----------              otherwise             |
  ||------             otherwise            | ||   /pi      \                                  |     ||--------              otherwise             | ||   /pi    \                                    |
  \\csc(x)                                  / ||csc|-- - 2*x|                                  |     \\csc(2*x)                                    / ||csc|-- - x|                                    |
                                              \\   \2       /                                  /                                                     \\   \2     /                                    /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{1}{\csc{\left(x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\frac{1}{\csc{\left(2 x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)$$
  //   0      for And(im(x) = 0, 3*x mod pi = 0)\
5*|<                                            |
  \\sin(3*x)              otherwise             /
$$5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod \pi = 0 \\\sin{\left(3 x \right)} & \text{otherwise} \end{cases}\right)$$
  //     0       for And(im(x) = 0, x mod pi = 0)\                                                                                                          //     1        for And(im(x) = 0, x mod 2*pi = 0)\
  ||                                             | //     1        for And(im(x) = 0, x mod pi = 0)\     //     0       for And(im(x) = 0, 2*x mod pi = 0)\ ||                                                |
  ||       /x\                                   | ||                                              |     ||                                               | ||        2/x\                                    |
  ||  2*cot|-|                                   | ||        2                                     |     ||  2*cot(x)                                     | ||-1 + cot |-|                                    |
5*|<       \2/                                   |*|<-1 + cot (x)                                  | + 5*|<-----------              otherwise             |*|<         \2/                                    |
  ||-----------             otherwise            | ||------------             otherwise            |     ||       2                                       | ||------------              otherwise             |
  ||       2/x\                                  | ||       2                                      |     ||1 + cot (x)                                    | ||       2/x\                                     |
  ||1 + cot |-|                                  | \\1 + cot (x)                                   /     \\                                               / ||1 + cot |-|                                     |
  \\        \2/                                  /                                                                                                          \\        \2/                                     /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
  //      0        for And(im(x) = 0, 3*x mod pi = 0)\
  ||                                                 |
  ||       /3*x\                                     |
  ||  2*cot|---|                                     |
5*|<       \ 2 /                                     |
  ||-------------              otherwise             |
  ||       2/3*x\                                    |
  ||1 + cot |---|                                    |
  \\        \ 2 /                                    /
$$5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 x}{2} \right)}}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
     /    pi\        /      pi\                         
5*cos|x - --|   5*cos|3*x - --|                         
     \    2 /        \      2 /                 /    pi\
------------- + --------------- + 5*cos(2*x)*cos|x - --|
      2                2                        \    2 /
$$5 \cos{\left(2 x \right)} \cos{\left(x - \frac{\pi}{2} \right)} + \frac{5 \cos{\left(x - \frac{\pi}{2} \right)}}{2} + \frac{5 \cos{\left(3 x - \frac{\pi}{2} \right)}}{2}$$
            /      pi\                 /    pi\
5*cos(x)*cos|2*x - --| + 5*cos(2*x)*cos|x - --|
            \      2 /                 \    2 /
$$5 \cos{\left(x \right)} \cos{\left(2 x - \frac{\pi}{2} \right)} + 5 \cos{\left(2 x \right)} \cos{\left(x - \frac{\pi}{2} \right)}$$
5*sin(3*x)
$$5 \sin{\left(3 x \right)}$$
            /pi      \                 /    pi\
5*sin(x)*sin|-- + 2*x| + 5*sin(2*x)*sin|x + --|
            \2       /                 \    2 /
$$5 \sin{\left(x \right)} \sin{\left(2 x + \frac{\pi}{2} \right)} + 5 \sin{\left(2 x \right)} \sin{\left(x + \frac{\pi}{2} \right)}$$
  //                    0                      for And(im(x) = 0, x mod pi = 0)\ //                     1                       for And(im(x) = 0, x mod pi = 0)\     //                      0                        for And(im(x) = 0, 2*x mod pi = 0)\ //                     1                       for And(im(x) = 0, x mod 2*pi = 0)\
  ||                                                                           | ||                                                                             |     ||                                                                                 | ||                                                                               |
5*|
$$\left(5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
       /3*x\ 
 10*tan|---| 
       \ 2 / 
-------------
       2/3*x\
1 + tan |---|
        \ 2 /
$$\frac{10 \tan{\left(\frac{3 x}{2} \right)}}{\tan^{2}{\left(\frac{3 x}{2} \right)} + 1}$$
  //  0     for And(im(x) = 0, x mod pi = 0)\     //   0      for And(im(x) = 0, 3*x mod pi = 0)\                                                                                              
5*|<                                        |   5*|<                                            |                                                                                              
  \\sin(x)             otherwise            /     \\sin(3*x)              otherwise             /     //  0     for And(im(x) = 0, x mod pi = 0)\ //   1      for And(im(x) = 0, x mod pi = 0)\
--------------------------------------------- + ------------------------------------------------- + 5*|<                                        |*|<                                          |
                      2                                                 2                             \\sin(x)             otherwise            / \\cos(2*x)             otherwise            /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\frac{5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right)}{2}\right) + \left(\frac{5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod \pi = 0 \\\sin{\left(3 x \right)} & \text{otherwise} \end{cases}\right)}{2}\right)$$
  //     0       for And(im(x) = 0, x mod pi = 0)\                                                   //      0        for And(im(x) = 0, 2*x mod pi = 0)\                                              
  ||                                             | //   1      for And(im(x) = 0, x mod pi = 0)\     ||                                                 | //  1     for And(im(x) = 0, x mod 2*pi = 0)\
  ||     1                                       | ||                                          |     ||      1                                          | ||                                          |
5*|<-----------             otherwise            |*|<   1                                      | + 5*|<-------------              otherwise             |*|<  1                                       |
  ||   /    pi\                                  | ||--------             otherwise            |     ||   /      pi\                                    | ||------              otherwise             |
  ||sec|x - --|                                  | \\sec(2*x)                                  /     ||sec|2*x - --|                                    | \\sec(x)                                    /
  \\   \    2 /                                  /                                                   \\   \      2 /                                    /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{1}{\sec{\left(2 x \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\frac{1}{\sec{\left(2 x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right)$$
  //  0     for And(im(x) = 0, x mod pi = 0)\ //   1      for And(im(x) = 0, x mod pi = 0)\     //   0      for And(im(x) = 0, 2*x mod pi = 0)\ //  1     for And(im(x) = 0, x mod 2*pi = 0)\
5*|<                                        |*|<                                          | + 5*|<                                            |*|<                                          |
  \\sin(x)             otherwise            / \\cos(2*x)             otherwise            /     \\sin(2*x)              otherwise             / \\cos(x)              otherwise             /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right)\right)$$
      5      
-------------
   /      pi\
sec|3*x - --|
   \      2 /
$$\frac{5}{\sec{\left(3 x - \frac{\pi}{2} \right)}}$$
5*sin(x)   5*sin(3*x)               /pi      \
-------- + ---------- + 5*sin(x)*sin|-- + 2*x|
   2           2                    \2       /
$$5 \sin{\left(x \right)} \sin{\left(2 x + \frac{\pi}{2} \right)} + \frac{5 \sin{\left(x \right)}}{2} + \frac{5 \sin{\left(3 x \right)}}{2}$$
      5                5                   5          
------------- + --------------- + --------------------
     /    pi\        /      pi\               /    pi\
2*sec|x - --|   2*sec|3*x - --|   sec(2*x)*sec|x - --|
     \    2 /        \      2 /               \    2 /
$$\frac{5}{2 \sec{\left(3 x - \frac{\pi}{2} \right)}} + \frac{5}{2 \sec{\left(x - \frac{\pi}{2} \right)}} + \frac{5}{\sec{\left(2 x \right)} \sec{\left(x - \frac{\pi}{2} \right)}}$$
  //                      0                         for And(im(x) = 0, x mod pi = 0)\                                                                                                                                                                                  //                        1                          for And(im(x) = 0, x mod 2*pi = 0)\
  ||                                                                                | //                       1                         for And(im(x) = 0, x mod pi = 0)\     //                       0                          for And(im(x) = 0, 2*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)                                    |
  |||                                                                               | ||/     1        for And(im(x) = 0, x mod pi = 0)                                  |     ||/     0       for And(im(x) = 0, 2*x mod pi = 0)                                    | |||                                                                                    |
  |||       /x\                                                                     | |||                                                                                |     |||                                                                                   | |||        2/x\                                                                        |
5*|<|  2*cot|-|                                                                     |*|<|        2                                                                       | + 5*|<|  2*cot(x)                                                                         |*|<|-1 + cot |-|                                                                        |
  ||<       \2/                                                otherwise            | ||<-1 + cot (x)                                               otherwise            |     ||<-----------              otherwise                           otherwise             | ||<         \2/                                                  otherwise             |
  |||-----------             otherwise                                              | |||------------             otherwise                                              |     |||       2                                                                           | |||------------              otherwise                                                 |
  |||       2/x\                                                                    | |||       2                                                                        |     |||1 + cot (x)                                                                        | |||       2/x\                                                                         |
  |||1 + cot |-|                                                                    | \\\1 + cot (x)                                                                     /     \\\                                                                                   / |||1 + cot |-|                                                                         |
  \\\        \2/                                                                    /                                                                                                                                                                                  \\\        \2/                                                                         /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
  //     0       for And(im(x) = 0, x mod pi = 0)\     //      0        for And(im(x) = 0, 3*x mod pi = 0)\                                                                                                       
  ||                                             |     ||                                                 |                                                                                                       
  ||       /x\                                   |     ||       /3*x\                                     |                                                                                                       
  ||  2*cot|-|                                   |     ||  2*cot|---|                                     |                                                                                                       
5*|<       \2/                                   |   5*|<       \ 2 /                                     |                                                                                                       
  ||-----------             otherwise            |     ||-------------              otherwise             |     //     0       for And(im(x) = 0, x mod pi = 0)\                                                  
  ||       2/x\                                  |     ||       2/3*x\                                    |     ||                                             | //     1        for And(im(x) = 0, x mod pi = 0)\
  ||1 + cot |-|                                  |     ||1 + cot |---|                                    |     ||       /x\                                   | ||                                              |
  \\        \2/                                  /     \\        \ 2 /                                    /     ||  2*cot|-|                                   | ||        2                                     |
-------------------------------------------------- + ------------------------------------------------------ + 5*|<       \2/                                   |*|<-1 + cot (x)                                  |
                        2                                                      2                                ||-----------             otherwise            | ||------------             otherwise            |
                                                                                                                ||       2/x\                                  | ||       2                                      |
                                                                                                                ||1 + cot |-|                                  | \\1 + cot (x)                                   /
                                                                                                                \\        \2/                                  /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\frac{5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)}{2}\right) + \left(\frac{5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 x}{2} \right)}}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)}{2}\right)$$
       /x\           /3*x\         /       2   \    /x\  
  5*tan|-|      5*tan|---|      10*\1 - tan (x)/*tan|-|  
       \2/           \ 2 /                          \2/  
----------- + ------------- + ---------------------------
       2/x\          2/3*x\   /       2   \ /       2/x\\
1 + tan |-|   1 + tan |---|   \1 + tan (x)/*|1 + tan |-||
        \2/           \ 2 /                 \        \2//
$$\frac{10 \cdot \left(1 - \tan^{2}{\left(x \right)}\right) \tan{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\tan^{2}{\left(x \right)} + 1\right)} + \frac{5 \tan{\left(\frac{3 x}{2} \right)}}{\tan^{2}{\left(\frac{3 x}{2} \right)} + 1} + \frac{5 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}$$
       5                 5       
--------------- + ---------------
csc(x)*sec(2*x)   csc(2*x)*sec(x)
$$\frac{5}{\csc{\left(2 x \right)} \sec{\left(x \right)}} + \frac{5}{\csc{\left(x \right)} \sec{\left(2 x \right)}}$$
                                              //      1        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, 2*x mod pi = 0)\ ||                                               |
5*|<                                        |*|<   /pi      \                                  | + 5*|<                                            |*|<   /    pi\                                    |
  \\sin(x)             otherwise            / ||sin|-- + 2*x|             otherwise            |     \\sin(2*x)              otherwise             / ||sin|x + --|              otherwise             |
                                              \\   \2       /                                  /                                                     \\   \    2 /                                    /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin{\left(2 x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\sin{\left(x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right)$$
Комбинаторика [src]
5*(cos(x)*sin(2*x) + cos(2*x)*sin(x))
$$5 \left(\sin{\left (x \right )} \cos{\left (2 x \right )} + \sin{\left (2 x \right )} \cos{\left (x \right )}\right)$$
Раскрыть выражение [src]
                  2          
-5*sin(x) + 20*cos (x)*sin(x)
$$20 \sin{\left(x \right)} \cos^{2}{\left(x \right)} - 5 \sin{\left(x \right)}$$