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Упрощение выражений/ tan(x)^3 если x=-1/2

tan(x)^3 если x=-1/2

Выражение, которое надо упростить:

Решение

Вы ввели [src] 
   3   
tan (x)
$$\tan^{3}{\left(x \right)}$$ 
tan(x)^3
Подстановка условия [src] 
tan(x)^3 при x = -1/2
подставляем
   3   
tan (x)
$$\tan^{3}{\left(x \right)}$$ 
   3   
tan (x)
$$\tan^{3}{\left(x \right)}$$ 
переменные
x = -1/2
$$x = - \frac{1}{2}$$ 
   3        
tan ((-1/2))
$$\tan^{3}{\left((-1/2) \right)}$$ 
   3      
tan (-1/2)
$$\tan^{3}{\left(- \frac{1}{2} \right)}$$ 
    3     
-tan (1/2)
$$- \tan^{3}{\left(\frac{1}{2} \right)}$$ 
-tan(1/2)^3
Степени [src] 
                   3 
   /   I*x    -I*x\  
-I*\- e    + e    /  
---------------------
                 3   
   / I*x    -I*x\    
   \e    + e    /
$$- \frac{i \left(- e^{i x} + e^{- i x}\right)^{3}}{\left(e^{i x} + e^{- i x}\right)^{3}}$$ 
-i*(-exp(i*x) + exp(-i*x))^3/(exp(i*x) + exp(-i*x))^3
Численный ответ [src] 
tan(x)^3
tan(x)^3
Тригонометрическая часть [src] 
   1   
-------
   3   
cot (x)
$$\frac{1}{\cot^{3}{\left(x \right)}}$$ 
   3   
sin (x)
-------
   3   
cos (x)
$$\frac{\sin^{3}{\left(x \right)}}{\cos^{3}{\left(x \right)}}$$ 
   3   
sec (x)
-------
   3   
csc (x)
$$\frac{\sec^{3}{\left(x \right)}}{\csc^{3}{\left(x \right)}}$$ 
     6   
8*sin (x)
---------
   3     
sin (2*x)
$$\frac{8 \sin^{6}{\left(x \right)}}{\sin^{3}{\left(2 x \right)}}$$ 
     3     
8*csc (2*x)
-----------
     6     
  csc (x)
$$\frac{8 \csc^{3}{\left(2 x \right)}}{\csc^{6}{\left(x \right)}}$$ 
     3      
  sin (x)   
------------
   3/    pi\
sin |x + --|
    \    2 /
$$\frac{\sin^{3}{\left(x \right)}}{\sin^{3}{\left(x + \frac{\pi}{2} \right)}}$$ 
   3/    pi\
cos |x - --|
    \    2 /
------------
     3      
  cos (x)
$$\frac{\cos^{3}{\left(x - \frac{\pi}{2} \right)}}{\cos^{3}{\left(x \right)}}$$ 
     3      
  sec (x)   
------------
   3/    pi\
sec |x - --|
    \    2 /
$$\frac{\sec^{3}{\left(x \right)}}{\sec^{3}{\left(x - \frac{\pi}{2} \right)}}$$ 
   3/pi    \
csc |-- - x|
    \2     /
------------
     3      
  csc (x)
$$\frac{\csc^{3}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{3}{\left(x \right)}}$$ 
     3      
  sec (x)   
------------
   3/pi    \
sec |-- - x|
    \2     /
$$\frac{\sec^{3}{\left(x \right)}}{\sec^{3}{\left(- x + \frac{\pi}{2} \right)}}$$ 
   3/pi    \
csc |-- - x|
    \2     /
------------
   3        
csc (pi - x)
$$\frac{\csc^{3}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{3}{\left(- x + \pi \right)}}$$ 
     3/      pi\
8*sec |2*x - --|
      \      2 /
----------------
     6/    pi\  
  sec |x - --|  
      \    2 /
$$\frac{8 \sec^{3}{\left(2 x - \frac{\pi}{2} \right)}}{\sec^{6}{\left(x - \frac{\pi}{2} \right)}}$$ 
     6/    pi\
8*cos |x - --|
      \    2 /
--------------
   3/      pi\
cos |2*x - --|
    \      2 /
$$\frac{8 \cos^{6}{\left(x - \frac{\pi}{2} \right)}}{\cos^{3}{\left(2 x - \frac{\pi}{2} \right)}}$$ 
       3/x\   
  8*tan |-|   
        \2/   
--------------
             3
/       2/x\\ 
|1 - tan |-|| 
\        \2//
$$\frac{8 \tan^{3}{\left(\frac{x}{2} \right)}}{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}}$$ 
-sin(3*x) + 3*sin(x)
--------------------
3*cos(x) + cos(3*x)
$$\frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{3 \cos{\left(x \right)} + \cos{\left(3 x \right)}}$$ 
      3/x\    3/x\
-8*cos |-|*sin |-|
       \2/     \2/
------------------
                3 
/          2/x\\  
|-1 + 2*sin |-||  
\           \2//
$$- \frac{8 \sin^{3}{\left(\frac{x}{2} \right)} \cos^{3}{\left(\frac{x}{2} \right)}}{\left(2 \sin^{2}{\left(\frac{x}{2} \right)} - 1\right)^{3}}$$ 
                3        
   /       2   \     6/x\
64*\1 + tan (x)/ *tan |-|
                      \2/
-------------------------
               6         
  /       2/x\\     3    
  |1 + tan |-|| *tan (x) 
  \        \2//
$$\frac{64 \left(\tan^{2}{\left(x \right)} + 1\right)^{3} \tan^{6}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6} \tan^{3}{\left(x \right)}}$$ 
                  3                
/       2/x   pi\\     3/x\    6/x\
|1 + tan |- + --|| *cot |-|*sin |-|
\        \2   4 //      \2/     \2/
-----------------------------------
               3/x   pi\           
            tan |- + --|           
                \2   4 /
$$\frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{3} \sin^{6}{\left(\frac{x}{2} \right)} \cot^{3}{\left(\frac{x}{2} \right)}}{\tan^{3}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}$$ 
                  3              
/       2/x   pi\\              3
|1 - cot |- + --|| *(1 + sin(x)) 
\        \2   4 //               
---------------------------------
                    3            
       /       2/x\\     6/x\    
     8*|1 - tan |-|| *cos |-|    
       \        \2//      \2/
$$\frac{\left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{3} \left(\sin{\left(x \right)} + 1\right)^{3}}{8 \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3} \cos^{6}{\left(\frac{x}{2} \right)}}$$ 
                  3        
/       2/x   pi\\     3/x\
|1 + tan |- + --|| *cot |-|
\        \2   4 //      \2/
---------------------------
             3             
/       2/x\\     3/x   pi\
|1 + cot |-|| *tan |- + --|
\        \2//      \2   4 /
$$\frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{3} \cot^{3}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3} \tan^{3}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}$$ 
             3                    3
/       2/x\\  /        2/x   pi\\ 
|1 + cot |-|| *|-1 + tan |- + --|| 
\        \2//  \         \2   4 // 
-----------------------------------
                  3               3
/       2/x   pi\\  /        2/x\\ 
|1 + tan |- + --|| *|-1 + cot |-|| 
\        \2   4 //  \         \2//
$$\frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{3} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{3} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{3}}$$ 
             3                   3
/       2/x\\  /       2/x   pi\\ 
|1 + tan |-|| *|1 - cot |- + --|| 
\        \2//  \        \2   4 // 
----------------------------------
                  3              3
/       2/x   pi\\  /       2/x\\ 
|1 + cot |- + --|| *|1 - tan |-|| 
\        \2   4 //  \        \2//
$$\frac{\left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{3} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}}{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3} \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{3}}$$ 
                               //   zoo     for 2*x mod pi = 0\
  //   0     for x mod pi = 0\ ||                             |
  ||                         | ||    3                        |
8*|<   6                     |*|< tan (x)                     |
  ||sin (x)     otherwise    | ||---------      otherwise     |
  \\                         / ||     6                       |
                               \\8*sin (x)                    /
$$8 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: 2 x \bmod \pi = 0 \\\frac{\tan^{3}{\left(x \right)}}{8 \sin^{6}{\left(x \right)}} & \text{otherwise} \end{cases}\right)$$ 
  //      0         for x mod pi = 0\                                      
  ||                                | //     zoo        for 2*x mod pi = 0\
  ||        6/x\                    | ||                                  |
  ||  64*cot |-|                    | ||             3                    |
  ||         \2/                    | ||/       2   \                     |
8*|<--------------     otherwise    |*|<\1 + cot (x)/                     |
  ||             6                  | ||--------------      otherwise     |
  ||/       2/x\\                   | ||       3                          |
  |||1 + cot |-||                   | ||  8*cot (x)                       |
  ||\        \2//                   | \\                                  /
  \\                                /
$$8 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{64 \cot^{6}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: 2 x \bmod \pi = 0 \\\frac{\left(\cot^{2}{\left(x \right)} + 1\right)^{3}}{8 \cot^{3}{\left(x \right)}} & \text{otherwise} \end{cases}\right)$$ 
                                               //                    /    3*pi\             \
                                               ||      1         for |x + ----| mod 2*pi = 0|
//           1             for x mod 2*pi = 0\ ||                    \     2  /             |
||                                           | ||                                           |
||           1                               | ||       3/x\                                |
||-----------------------      otherwise     | ||  8*tan |-|                                |
|<              3                            |*|<        \2/                                |
||/        2/x\\     6/x\                    | ||--------------           otherwise         |
|||-1 + cot |-|| *sin |-|                    | ||             3                             |
||\         \2//      \2/                    | ||/       2/x\\                              |
\\                                           / |||1 + tan |-||                              |
                                               ||\        \2//                              |
                                               \\                                           /
$$\left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{3} \sin^{6}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{8 \tan^{3}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}} & \text{otherwise} \end{cases}\right)$$ 
                                       //                          /    3*pi\             \
//       1         for x mod 2*pi = 0\ ||         1            for |x + ----| mod 2*pi = 0|
||                                   | ||                          \     2  /             |
||              3                    | ||                                                 |
|| /       2/x\\                     | ||                   3                             |
|| |1 + cot |-||                     | ||/        2/x   pi\\                              |
|< \        \2//                     |*|<|-1 + tan |- + --||                              |
||---------------      otherwise     | ||\         \2   4 //                              |
||              3                    | ||--------------------           otherwise         |
||/        2/x\\                     | ||                  3                              |
|||-1 + cot |-||                     | ||/       2/x   pi\\                               |
\\\         \2//                     / |||1 + tan |- + --||                               |
                                       \\\        \2   4 //                               /
$$\left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{3}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{3}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{3}} & \text{otherwise} \end{cases}\right)$$ 
Piecewise((1, Mod(x = 2*pi, 0)), ((1 + cot(x/2)^2)^3/(-1 + cot(x/2)^2)^3, True))*Piecewise((1, Mod(x + 3*pi/2 = 2*pi, 0)), ((-1 + tan(x/2 + pi/4)^2)^3/(1 + tan(x/2 + pi/4)^2)^3, True))
    © Господин Экзамен – Калькулятор