Господин Экзамен

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Упрощение выражений/ tan(x)-x если x=-3

tan(x)-x если x=-3

Выражение, которое надо упростить:

Решение

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