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sin(pi-t)*cos(2*pi-t)/tan(pi-t)*cos(pi-t)еслиt=2 (упростите выражение)

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Примеры

Решение

Вы ввели [src]
sin(pi - t)*cos(2*pi - t)*cos(pi - t)
-------------------------------------
             tan(pi - t)
$$\frac{\sin{\left(\pi - t \right)} \cos{\left(\pi - t \right)} \cos{\left(- t + 2 \pi \right)}}{\tan{\left(\pi - t \right)}}$$
Подстановка условия [src]
sin(pi - t)*cos(2*pi - t)*cos(pi - t)/tan(pi - t) при t = 2
подставляем
sin(pi - t)*cos(2*pi - t)*cos(pi - t)
-------------------------------------
             tan(pi - t)
$$\frac{\sin{\left(\pi - t \right)} \cos{\left(\pi - t \right)} \cos{\left(- t + 2 \pi \right)}}{\tan{\left(\pi - t \right)}}$$
   3   
cos (t)
$$\cos^{3}{\left(t \right)}$$
переменные
t = 2
$$t = 2$$
   3     
cos ((2))
$$\cos^{3}{\left((2) \right)}$$
   3   
cos (2)
$$\cos^{3}{\left(2 \right)}$$
Степени [src]
   2          
cos (t)*sin(t)
--------------
    tan(t)
$$\frac{\sin{\left(t \right)} \cos^{2}{\left(t \right)}}{\tan{\left(t \right)}}$$
 / I*(pi - t)    I*(t - pi)\ / I*(t - 2*pi)    I*(-t + 2*pi)\                                                           
 |e             e          | |e               e             | /   I*(t - pi)    I*(pi - t)\ / I*(pi - t)    I*(t - pi)\ 
-|----------- + -----------|*|------------- + --------------|*\- e           + e          /*\e           + e          / 
 \     2             2     / \      2               2       /                                                           
------------------------------------------------------------------------------------------------------------------------
                                              /   I*(pi - t)    I*(t - pi)\                                             
                                            2*\- e           + e          /
$$- \frac{\left(\frac{e^{i \left(\pi - t\right)}}{2} + \frac{e^{i \left(t - \pi\right)}}{2}\right) \left(e^{i \left(\pi - t\right)} - e^{i \left(t - \pi\right)}\right) \left(e^{i \left(\pi - t\right)} + e^{i \left(t - \pi\right)}\right) \left(\frac{e^{i \left(- t + 2 \pi\right)}}{2} + \frac{e^{i \left(t - 2 \pi\right)}}{2}\right)}{2 \left(- e^{i \left(\pi - t\right)} + e^{i \left(t - \pi\right)}\right)}$$
Численный ответ [src]
cos(pi - t)*cos(2*pi - t)*sin(pi - t)/tan(pi - t)
Рациональный знаменатель [src]
   2          
cos (t)*sin(t)
--------------
    tan(t)
$$\frac{\sin{\left(t \right)} \cos^{2}{\left(t \right)}}{\tan{\left(t \right)}}$$
Объединение рациональных выражений [src]
   2          
cos (t)*sin(t)
--------------
    tan(t)
$$\frac{\sin{\left(t \right)} \cos^{2}{\left(t \right)}}{\tan{\left(t \right)}}$$
Общее упрощение [src]
   3   
cos (t)
$$\cos^{3}{\left(t \right)}$$
Собрать выражение [src]
cos(3*t)   3*cos(t)
-------- + --------
   4          4
$$\frac{3}{4} \cos{\left (t \right )} + \frac{1}{4} \cos{\left (3 t \right )}$$
cos(pi - t)*cos(2*pi - t)*sin(pi - t)
-------------------------------------
             tan(pi - t)
$$\frac{\sin{\left (- t + \pi \right )} \cos{\left (- t + \pi \right )}}{\tan{\left (- t + \pi \right )}} \cos{\left (- t + 2 \pi \right )}$$
Общий знаменатель [src]
   2          
cos (t)*sin(t)
--------------
    tan(t)
$$\frac{\sin{\left(t \right)} \cos^{2}{\left(t \right)}}{\tan{\left(t \right)}}$$
Тригонометрическая часть [src]
      1                 csc(t)        
------------- + ----------------------
     /pi    \                 /pi    \
4*csc|-- - t|   4*csc(3*t)*csc|-- - t|
     \2     /                 \2     /
$$\frac{\csc{\left(t \right)}}{4 \csc{\left(3 t \right)} \csc{\left(- t + \frac{\pi}{2} \right)}} + \frac{1}{4 \csc{\left(- t + \frac{\pi}{2} \right)}}$$
/  1     for And(im(t) = 0, t mod 2*pi = 0)   //   0      for And(im(t) = 0, 3*t mod pi = 0)\       
<                                             |<                                            |*cot(t)
\cos(t)              otherwise                \\sin(3*t)              otherwise             /       
------------------------------------------- + ------------------------------------------------------
                     4                                                  4
$$\left(\frac{\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge 3 t \bmod \pi = 0 \\\sin{\left(3 t \right)} & \text{otherwise} \end{cases}\right) \cot{\left(t \right)}}{4}\right) + \left(\frac{\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge t \bmod 2 \pi = 0 \\\cos{\left(t \right)} & \text{otherwise} \end{cases}}{4}\right)$$
             3
/       2/t\\ 
|1 - tan |-|| 
\        \2// 
--------------
             3
/       2/t\\ 
|1 + tan |-|| 
\        \2//
$$\frac{\left(1 - \tan^{2}{\left(\frac{t}{2} \right)}\right)^{3}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{3}}$$
   1   
-------
   3   
sec (t)
$$\frac{1}{\sec^{3}{\left(t \right)}}$$
                                                                                                                     //     1        for And(im(t) = 0, (pi - t) mod 2*pi = 0)\       
                                                                                                                     ||                                                       |       
 //         0            for And(im(t) = 0, (-t) mod pi = 0)\ //     1        for And(im(t) = 0, (-t) mod 2*pi = 0)\ ||        1                                              |       
 ||                                                         | ||                                                   | ||-1 + -------                                           |       
 ||         2                                               | ||        2/t\                                       | ||        2/t\                                           |       
 ||--------------------               otherwise             | ||-1 + cot |-|                                       | ||     cot |-|                                           |       
-|
$$- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge - t \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}\right) \cot{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge - t \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{t}{2} \right)} - 1}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge \left(\pi - t\right) \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}} & \text{otherwise} \end{cases}\right) \cot{\left(t \right)}$$
   /    pi\                    
sin|t + --|                    
   \    2 /   sin(2*t)*sin(3*t)
----------- + -----------------
     4                 2       
                  8*sin (t)
$$\frac{\sin{\left(t + \frac{\pi}{2} \right)}}{4} + \frac{\sin{\left(2 t \right)} \sin{\left(3 t \right)}}{8 \sin^{2}{\left(t \right)}}$$
                                                //     1       for And(im(t) = 0, (-t) mod 2*pi = 0)\ //     1        for And(im(t) = 0, (pi - t) mod 2*pi = 0)\          
 //  0     for And(im(t) = 0, (-t) mod pi = 0)\ ||                                                  | ||                                                       |          
-|<                                           |*|<   /    pi\                                       |*|<    /    pi\                                           |*sin(2*t) 
 \\sin(t)               otherwise             / ||sin|t + --|                otherwise              | ||-sin|t + --|                  otherwise                |          
                                                \\   \    2 /                                       / \\    \    2 /                                           /          
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                     2                                                                                    
                                                                                2*sin (t)
$$- \frac{\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge - t \bmod \pi = 0 \\\sin{\left(t \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge - t \bmod 2 \pi = 0 \\\sin{\left(t + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge \left(\pi - t\right) \bmod 2 \pi = 0 \\- \sin{\left(t + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \sin{\left(2 t \right)}}{2 \sin^{2}{\left(t \right)}}$$
 //                    0                      for And(im(t) = 0, (-t) mod pi = 0)\ //                     1                       for And(im(t) = 0, (-t) mod 2*pi = 0)\ //                      1                         for And(im(t) = 0, (pi - t) mod 2*pi = 0)\       
 ||                                                                              | ||                                                                                  | ||                                                                                         |       
-|
$$- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge - t \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge t \bmod \pi = 0 \\\sin{\left(t \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge - t \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge t \bmod 2 \pi = 0 \\\cos{\left(t \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge \left(\pi - t\right) \bmod 2 \pi = 0 \\- \begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge t \bmod 2 \pi = 0 \\\cos{\left(t \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \cot{\left(t \right)}$$
 //  0     for And(im(t) = 0, (-t) mod pi = 0)\ //  1     for And(im(t) = 0, (-t) mod 2*pi = 0)\ //   1     for And(im(t) = 0, (pi - t) mod 2*pi = 0)\       
-|<                                           |*|<                                             |*|<                                                  |*cot(t)
 \\sin(t)               otherwise             / \\cos(t)                otherwise              / \\-cos(t)                  otherwise                /
$$- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge - t \bmod \pi = 0 \\\sin{\left(t \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge - t \bmod 2 \pi = 0 \\\cos{\left(t \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge \left(\pi - t\right) \bmod 2 \pi = 0 \\- \cos{\left(t \right)} & \text{otherwise} \end{cases}\right) \cot{\left(t \right)}$$
/           1             for And(im(t) = 0, t mod 2*pi = 0)
|                                                           
|              3
$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge t \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{3} \sin^{6}{\left(\frac{t}{2} \right)} & \text{otherwise} \end{cases}$$
                          2              2       
            2 /       /t\\  /        /t\\        
(1 + cos(t)) *|1 + tan|-|| *|-1 + tan|-|| *cos(t)
              \       \2//  \        \2//        
-------------------------------------------------
                        4
$$\frac{\left(\cos{\left(t \right)} + 1\right)^{2} \left(\tan{\left(\frac{t}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{t}{2} \right)} + 1\right)^{2} \cos{\left(t \right)}}{4}$$
         /       1   \ /       2/t\\      
      -2*|1 - -------|*|1 - tan |-||      
         |       2/t\| \        \2//      
         |    tan |-||                    
         \        \2//                    
------------------------------------------
             2                            
/       1   \  /       2/t\\           /t\
|1 + -------| *|1 + tan |-||*tan(t)*tan|-|
|       2/t\|  \        \2//           \2/
|    tan |-||                             
\        \2//
$$- \frac{2 \cdot \left(1 - \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}\right) \left(1 - \tan^{2}{\left(\frac{t}{2} \right)}\right)}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right) \tan{\left(\frac{t}{2} \right)} \tan{\left(t \right)}}$$
 //     0       for And(im(t) = 0, (-t) mod pi = 0)\                                                                                                                   
 ||                                                | //  1     for And(im(t) = 0, (-t) mod 2*pi = 0)\ //  1     for And(im(t) = 0, (pi - t) mod 2*pi = 0)\             
 ||     1                                          | ||                                             | ||                                                 |    /    pi\ 
-|<-----------               otherwise             |*|<  1                                          |*|< -1                                              |*sec|t - --| 
 ||   /    pi\                                     | ||------                otherwise              | ||------                  otherwise                |    \    2 / 
 ||sec|t - --|                                     | \\sec(t)                                       / \\sec(t)                                           /             
 \\   \    2 /                                     /                                                                                                                   
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                 sec(t)
$$- \frac{\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge - t \bmod \pi = 0 \\\frac{1}{\sec{\left(t - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge - t \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(t \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge \left(\pi - t\right) \bmod 2 \pi = 0 \\- \frac{1}{\sec{\left(t \right)}} & \text{otherwise} \end{cases}\right) \sec{\left(t - \frac{\pi}{2} \right)}}{\sec{\left(t \right)}}$$
             /     3*pi\    /     5*pi\ 
-sin(2*t)*sin|-t + ----|*sin|-t + ----| 
             \      2  /    \      2  / 
----------------------------------------
                2*sin(t)
$$- \frac{\sin{\left(2 t \right)} \sin{\left(- t + \frac{3 \pi}{2} \right)} \sin{\left(- t + \frac{5 \pi}{2} \right)}}{2 \sin{\left(t \right)}}$$
   /       2/t\\ /       2/t\\    /t\
-2*|1 - cot |-||*|1 - tan |-||*cot|-|
   \        \2// \        \2//    \2/
-------------------------------------
              2                      
 /       2/t\\  /       2/t\\        
 |1 + cot |-|| *|1 + tan |-||*tan(t) 
 \        \2//  \        \2//
$$- \frac{2 \cdot \left(1 - \tan^{2}{\left(\frac{t}{2} \right)}\right) \left(1 - \cot^{2}{\left(\frac{t}{2} \right)}\right) \cot{\left(\frac{t}{2} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2} \tan{\left(t \right)}}$$
   2            
cos (t)*sin(2*t)
----------------
    2*sin(t)
$$\frac{\sin{\left(2 t \right)} \cos^{2}{\left(t \right)}}{2 \sin{\left(t \right)}}$$
 //     0       for And(im(t) = 0, (-t) mod pi = 0)\ //     1       for And(im(t) = 0, (-t) mod 2*pi = 0)\ //       1         for And(im(t) = 0, (pi - t) mod 2*pi = 0)\ 
 ||                                                | ||                                                  | ||                                                          | 
 ||       /t\                                      | ||       2/t\                                       | || /       2/t\\                                            | 
 ||  2*tan|-|                                      | ||1 - tan |-|                                       | ||-|1 - tan |-||                                            | 
-|<       \2/                                      |*|<        \2/                                       |*|< \        \2//                                            | 
 ||-----------               otherwise             | ||-----------                otherwise              | ||---------------                  otherwise                | 
 ||       2/t\                                     | ||       2/t\                                       | ||         2/t\                                             | 
 ||1 + tan |-|                                     | ||1 + tan |-|                                       | ||  1 + tan |-|                                             | 
 \\        \2/                                     / \\        \2/                                       / \\          \2/                                             / 
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                  tan(t)
$$- \frac{\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge - t \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{t}{2} \right)}}{\tan^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge - t \bmod 2 \pi = 0 \\\frac{1 - \tan^{2}{\left(\frac{t}{2} \right)}}{\tan^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge \left(\pi - t\right) \bmod 2 \pi = 0 \\- \frac{1 - \tan^{2}{\left(\frac{t}{2} \right)}}{\tan^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)}{\tan{\left(t \right)}}$$
 //     0       for And(im(t) = 0, (-t) mod pi = 0)\                                                                                                               
 ||                                                | //  1     for And(im(t) = 0, (-t) mod 2*pi = 0)\ //   1     for And(im(t) = 0, (pi - t) mod 2*pi = 0)\        
-|<   /    pi\                                     |*|<                                             |*|<                                                  |*cos(t) 
 ||cos|t - --|               otherwise             | \\cos(t)                otherwise              / \\-cos(t)                  otherwise                /        
 \\   \    2 /                                     /                                                                                                               
-------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                               /    pi\                                                                            
                                                                            cos|t - --|                                                                            
                                                                               \    2 /
$$- \frac{\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge - t \bmod \pi = 0 \\\cos{\left(t - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge - t \bmod 2 \pi = 0 \\\cos{\left(t \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge \left(\pi - t\right) \bmod 2 \pi = 0 \\- \cos{\left(t \right)} & \text{otherwise} \end{cases}\right) \cos{\left(t \right)}}{\cos{\left(t - \frac{\pi}{2} \right)}}$$
 //     0       for And(im(t) = 0, (-t) mod pi = 0)\ //     1        for And(im(t) = 0, (-t) mod 2*pi = 0)\ //     1        for And(im(t) = 0, (pi - t) mod 2*pi = 0)\       
 ||                                                | ||                                                   | ||                                                       |       
 ||       /t\                                      | ||        2/t\                                       | ||        2/t\                                           |       
 ||  2*tan|-|                                      | ||-1 + cot |-|                                       | ||-1 + tan |-|                                           |       
-|<       \2/                                      |*|<         \2/                                       |*|<         \2/                                           |*cot(t)
 ||-----------               otherwise             | ||------------                otherwise              | ||------------                  otherwise                |       
 ||       2/t\                                     | ||       2/t\                                        | ||       2/t\                                            |       
 ||1 + tan |-|                                     | ||1 + cot |-|                                        | ||1 + tan |-|                                            |       
 \\        \2/                                     / \\        \2/                                        / \\        \2/                                            /
$$- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge - t \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{t}{2} \right)}}{\tan^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge - t \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{t}{2} \right)} - 1}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge \left(\pi - t\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{t}{2} \right)} - 1}{\tan^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \cot{\left(t \right)}$$
       /    pi\    
    sec|t - --|    
       \    2 /    
-------------------
   3       /pi    \
sec (t)*sec|-- - t|
           \2     /
$$\frac{\sec{\left(t - \frac{\pi}{2} \right)}}{\sec^{3}{\left(t \right)} \sec{\left(- t + \frac{\pi}{2} \right)}}$$
         2/t\                /3*t\        
  1 - tan |-|             tan|---|        
          \2/                \ 2 /        
--------------- + ------------------------
  /       2/t\\     /       2/3*t\\       
4*|1 + tan |-||   2*|1 + tan |---||*tan(t)
  \        \2//     \        \ 2 //
$$\frac{1 - \tan^{2}{\left(\frac{t}{2} \right)}}{4 \left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)} + \frac{\tan{\left(\frac{3 t}{2} \right)}}{2 \left(\tan^{2}{\left(\frac{3 t}{2} \right)} + 1\right) \tan{\left(t \right)}}$$
                   /      pi\
         cos(t)*cos|3*t - --|
cos(t)             \      2 /
------ + --------------------
  4              /    pi\    
            4*cos|t - --|    
                 \    2 /
$$\frac{\cos{\left(t \right)}}{4} + \frac{\cos{\left(t \right)} \cos{\left(3 t - \frac{\pi}{2} \right)}}{4 \cos{\left(t - \frac{\pi}{2} \right)}}$$
/       1         for And(im(t) = 0, t mod 2*pi = 0)
|                                                   
|              3                                    
|/        2/t\\                                     
||-1 + cot |-||                                     
<\         \2//                                     
|---------------              otherwise             
|              3                                    
| /       2/t\\                                     
| |1 + cot |-||                                     
\ \        \2//
$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge t \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{3}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{3}} & \text{otherwise} \end{cases}$$
 //                      0                         for And(im(t) = 0, (-t) mod pi = 0)\ //                        1                          for And(im(t) = 0, (-t) mod 2*pi = 0)\ //                         1                            for And(im(t) = 0, (pi - t) mod 2*pi = 0)\       
 ||                                                                                   | ||                                                                                        | ||                                                                                               |       
 ||/     0       for And(im(t) = 0, t mod pi = 0)                                     | ||/     1        for And(im(t) = 0, t mod 2*pi = 0)                                       | || //     1        for And(im(t) = 0, t mod 2*pi = 0)\                                           |       
 |||                                                                                  | |||                                                                                       | || ||                                                |                                           |       
 |||       /t\                                                                        | |||        2/t\                                                                           | || ||        2/t\                                    |                                           |       
-|<|  2*cot|-|                                                                        |*|<|-1 + cot |-|                                                                           |*|< ||-1 + cot |-|                                    |                                           |*cot(t)
 ||<       \2/                                                  otherwise             | ||<         \2/                                                    otherwise              | ||-|<         \2/                                    |                  otherwise                |       
 |||-----------             otherwise                                                 | |||------------              otherwise                                                    | || ||------------              otherwise             |                                           |       
 |||       2/t\                                                                       | |||       2/t\                                                                            | || ||       2/t\                                     |                                           |       
 |||1 + cot |-|                                                                       | |||1 + cot |-|                                                                            | || ||1 + cot |-|                                     |                                           |       
 \\\        \2/                                                                       / \\\        \2/                                                                            / \\ \\        \2/                                     /                                           /
$$- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge - t \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge t \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{t}{2} \right)}}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge - t \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge t \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{t}{2} \right)} - 1}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge \left(\pi - t\right) \bmod 2 \pi = 0 \\- \begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge t \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{t}{2} \right)} - 1}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \cot{\left(t \right)}$$
cos(t)   sin(3*t)
------ + --------
  4      4*tan(t)
$$\frac{\sin{\left(3 t \right)}}{4 \tan{\left(t \right)}} + \frac{\cos{\left(t \right)}}{4}$$
 //  0     for And(im(t) = 0, (-t) mod pi = 0)\ //  1     for And(im(t) = 0, (-t) mod 2*pi = 0)\ //   1     for And(im(t) = 0, (pi - t) mod 2*pi = 0)\ 
-|<                                           |*|<                                             |*|<                                                  | 
 \\sin(t)               otherwise             / \\cos(t)                otherwise              / \\-cos(t)                  otherwise                / 
-------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                         tan(t)
$$- \frac{\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge - t \bmod \pi = 0 \\\sin{\left(t \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge - t \bmod 2 \pi = 0 \\\cos{\left(t \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge \left(\pi - t\right) \bmod 2 \pi = 0 \\- \cos{\left(t \right)} & \text{otherwise} \end{cases}\right)}{\tan{\left(t \right)}}$$
                                                //     1       for And(im(t) = 0, (-t) mod 2*pi = 0)\ //     1       for And(im(t) = 0, (pi - t) mod 2*pi = 0)\        
 //  0     for And(im(t) = 0, (-t) mod pi = 0)\ ||                                                  | ||                                                      |        
 ||                                           | ||     1                                            | ||    -1                                                |        
-|<  1                                        |*|<-----------                otherwise              |*|<-----------                  otherwise                |*csc(t) 
 ||------               otherwise             | ||   /pi    \                                       | ||   /pi    \                                           |        
 \\csc(t)                                     / ||csc|-- - t|                                       | ||csc|-- - t|                                           |        
                                                \\   \2     /                                       / \\   \2     /                                           /        
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                 /pi    \                                                                              
                                                                              csc|-- - t|                                                                              
                                                                                 \2     /
$$- \frac{\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge - t \bmod \pi = 0 \\\frac{1}{\csc{\left(t \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge - t \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- t + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge \left(\pi - t\right) \bmod 2 \pi = 0 \\- \frac{1}{\csc{\left(- t + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \csc{\left(t \right)}}{\csc{\left(- t + \frac{\pi}{2} \right)}}$$
   3   
cos (t)
$$\cos^{3}{\left(t \right)}$$
     1      
------------
   3/pi    \
csc |-- - t|
    \2     /
$$\frac{1}{\csc^{3}{\left(- t + \frac{\pi}{2} \right)}}$$
             3        
/       2/t\\     6/t\
|1 - tan |-|| *cos |-|
\        \2//      \2/
$$\left(1 - \tan^{2}{\left(\frac{t}{2} \right)}\right)^{3} \cos^{6}{\left(\frac{t}{2} \right)}$$
/     1        for And(im(t) = 0, t mod 2*pi = 0)   //      0        for And(im(t) = 0, 3*t mod pi = 0)\       
|                                                   ||                                                 |       
|        2/t\                                       ||       /3*t\                                     |       
|-1 + cot |-|                                       ||  2*cot|---|                                     |       
<         \2/                                       |<       \ 2 /                                     |*cot(t)
|------------              otherwise                ||-------------              otherwise             |       
|       2/t\                                        ||       2/3*t\                                    |       
|1 + cot |-|                                        ||1 + cot |---|                                    |       
\        \2/                                        \\        \ 2 /                                    /       
------------------------------------------------- + -----------------------------------------------------------
                        4                                                        4
$$\left(\frac{\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge 3 t \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 t}{2} \right)}}{\cot^{2}{\left(\frac{3 t}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \cot{\left(t \right)}}{4}\right) + \left(\frac{\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge t \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{t}{2} \right)} - 1}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases}}{4}\right)$$
   3/    pi\
sin |t + --|
    \    2 /
$$\sin^{3}{\left(t + \frac{\pi}{2} \right)}$$
                   /    pi\      
                sec|t - --|      
   1               \    2 /      
-------- + ----------------------
4*sec(t)               /      pi\
           4*sec(t)*sec|3*t - --|
                       \      2 /
$$\frac{\sec{\left(t - \frac{\pi}{2} \right)}}{4 \sec{\left(t \right)} \sec{\left(3 t - \frac{\pi}{2} \right)}} + \frac{1}{4 \sec{\left(t \right)}}$$
Комбинаторика [src]
   2          
cos (t)*sin(t)
--------------
    tan(t)
$$\frac{\sin{\left(t \right)} \cos^{2}{\left(t \right)}}{\tan{\left(t \right)}}$$
Раскрыть выражение [src]
   2          
cos (t)*sin(t)
--------------
    tan(t)
$$\frac{\sin{\left(t \right)} \cos^{2}{\left(t \right)}}{\tan{\left(t \right)}}$$