-8*sin(x)^2+5-8*cos(x)^2еслиx=2 (упростите выражение)

Вы ввели

       2               2   
- 8*sin (x) + 5 - 8*cos (x)

$$- 8 \sin^{2}{\left(x \right)} - 8 \cos^{2}{\left(x \right)} + 5$$

Подстановка условия

-8*sin(x)^2 + 5 - 8*cos(x)^2 при x = 2

-8*sin(x)^2 + 5 - 8*cos(x)^2

$$- 8 \sin^{2}{\left (x \right )} + 5 - 8 \cos^{2}{\left (x \right )}$$

-8*sin((2))^2 + 5 - 8*cos((2))^2

$$- 8 \sin^{2}{\left ((2) \right )} + 5 - 8 \cos^{2}{\left ((2) \right )}$$

-8*sin(2)^2 + 5 - 8*cos(2)^2

$$- 8 \sin^{2}{\left (2 \right )} + 5 - 8 \cos^{2}{\left (2 \right )}$$

5 - 8*cos(2)^2 - 8*sin(2)^2

$$- 8 \sin^{2}{\left (2 \right )} - 8 \cos^{2}{\left (2 \right )} + 5$$

Степени

                    2                      
      / I*x    -I*x\                      2
      |e      e    |      /   -I*x    I*x\ 
5 - 8*|---- + -----|  + 2*\- e     + e   / 
      \ 2       2  /                       

$$- 8 \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2} + 2 \left(e^{i x} - e^{- i x}\right)^{2} + 5$$

Численный ответ

5.0 - 8.0*cos(x)^2 - 8.0*sin(x)^2

Рациональный знаменатель

         2           2   
5 - 8*cos (x) - 8*sin (x)

$$- 8 \sin^{2}{\left (x \right )} - 8 \cos^{2}{\left (x \right )} + 5$$

Объединение рациональных выражений

         2           2   
5 - 8*cos (x) - 8*sin (x)

$$- 8 \sin^{2}{\left (x \right )} - 8 \cos^{2}{\left (x \right )} + 5$$

Общее упрощение

-3

$$-3$$

Собрать выражение

-3

$$-3$$

         2           2   
5 - 8*sin (x) - 8*cos (x)

$$- 8 \sin^{2}{\left (x \right )} - 8 \cos^{2}{\left (x \right )} + 5$$

Общий знаменатель

         2           2   
5 - 8*cos (x) - 8*sin (x)

$$- 8 \sin^{2}{\left (x \right )} - 8 \cos^{2}{\left (x \right )} + 5$$

Тригонометрическая часть

       8           8      
5 - ------- - ------------
       2         2/pi    \
    csc (x)   csc |-- - x|
                  \2     /

$$5 - \frac{8}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} - \frac{8}{\csc^{2}{\left(x \right)}}$$

      //   0     for And(im(x) = 0, x mod pi = 0)\     //     1        for And(im(x) = 0, x mod 2*pi = 0)\
      ||                                         |     ||                                                |
5 - 8*|<   2                                     | - 8*|<   2/    pi\                                    |
      ||sin (x)             otherwise            |     ||sin |x + --|              otherwise             |
      \\                                         /     \\    \    2 /                                    /

$$\left(- 8 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) - \left(8 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\sin^{2}{\left(x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + 5$$

      //     0        for And(im(x) = 0, x mod pi = 0)\     //   1     for And(im(x) = 0, x mod 2*pi = 0)\
      ||                                              |     ||                                           |
      ||     1                                        |     ||   1                                       |
5 - 8*|<------------             otherwise            | - 8*|<-------              otherwise             |
      ||   2/    pi\                                  |     ||   2                                       |
      ||sec |x - --|                                  |     ||sec (x)                                    |
      \\    \    2 /                                  /     \\                                           /

$$\left(- 8 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(8 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right) + 5$$

       8           8      
5 - ------- - ------------
       2         2/    pi\
    sec (x)   sec |x - --|
                  \    2 /

$$5 - \frac{8}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} - \frac{8}{\sec^{2}{\left(x \right)}}$$

      //                    0                       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)                                  |     ||/   1     for And(im(x) = 0, x mod 2*pi = 0)                                    |
5 - 8*|<|                                                                           | - 8*|<|                                                                               |
      ||<   2                                                  otherwise            |     ||<   2                                                     otherwise             |
      |||sin (x)             otherwise                                              |     |||cos (x)              otherwise                                                 |
      \\\                                                                           /     \\\                                                                               /

$$\left(- 8 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - \left(8 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + 5$$

       8         8   
5 - ------- - -------
       2         2   
    csc (x)   sec (x)

$$5 - \frac{8}{\sec^{2}{\left(x \right)}} - \frac{8}{\csc^{2}{\left(x \right)}}$$

         2           2/    pi\
5 - 8*cos (x) - 8*cos |x - --|
                      \    2 /

$$- 8 \cos^{2}{\left(x \right)} - 8 \cos^{2}{\left(x - \frac{\pi}{2} \right)} + 5$$

      //   0     for And(im(x) = 0, x mod pi = 0)\     //     1        for And(im(x) = 0, x mod 2*pi = 0)\
      ||                                         |     ||                                                |
      ||   1                                     |     ||     1                                          |
5 - 8*|<-------             otherwise            | - 8*|<------------              otherwise             |
      ||   2                                     |     ||   2/pi    \                                    |
      ||csc (x)                                  |     ||csc |-- - x|                                    |
      \\                                         /     \\    \2     /                                    /

$$\left(- 8 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(8 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + 5$$

                                    2
            2/x\       /       2/x\\ 
      32*tan |-|     8*|1 - tan |-|| 
             \2/       \        \2// 
5 - -------------- - ----------------
                 2                 2 
    /       2/x\\     /       2/x\\  
    |1 + tan |-||     |1 + tan |-||  
    \        \2//     \        \2//  

$$- \frac{8 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 5 - \frac{32 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$

         2           2/    pi\
5 - 8*sin (x) - 8*sin |x + --|
                      \    2 /

$$- 8 \sin^{2}{\left(x \right)} - 8 \sin^{2}{\left(x + \frac{\pi}{2} \right)} + 5$$

      //   0     for And(im(x) = 0, x mod pi = 0)\     //   1     for And(im(x) = 0, x mod 2*pi = 0)\
      ||                                         |     ||                                           |
5 - 8*|<   2                                     | - 8*|<   2                                       |
      ||sin (x)             otherwise            |     ||cos (x)              otherwise             |
      \\                                         /     \\                                           /

$$\left(- 8 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) - \left(8 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + 5$$

      //                        0                          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)                                  |     ||/       1         for And(im(x) = 0, x mod 2*pi = 0)                                    |
      |||                                                                                  |     |||                                                                                       |
      |||       2/x\                                                                       |     |||              2                                                                        |
      |||  4*cot |-|                                                                       |     |||/        2/x\\                                                                         |
5 - 8*|<|        \2/                                                                       | - 8*|<||-1 + cot |-||                                                                         |
      ||<--------------             otherwise                         otherwise            |     ||<\         \2//                                                   otherwise             |
      |||             2                                                                    |     |||---------------              otherwise                                                 |
      |||/       2/x\\                                                                     |     |||              2                                                                        |
      ||||1 + cot |-||                                                                     |     ||| /       2/x\\                                                                         |
      |||\        \2//                                                                     |     ||| |1 + cot |-||                                                                         |
      \\\                                                                                  /     \\\ \        \2//                                                                         /

$$\left(- 8 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - \left(8 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + 5$$

-3

$$-3$$

      //      0         for And(im(x) = 0, x mod pi = 0)\     //       1         for And(im(x) = 0, x mod 2*pi = 0)\
      ||                                                |     ||                                                   |
      ||       2/x\                                     |     ||              2                                    |
      ||  4*cot |-|                                     |     ||/        2/x\\                                     |
      ||        \2/                                     |     |||-1 + cot |-||                                     |
5 - 8*|<--------------             otherwise            | - 8*|<\         \2//                                     |
      ||             2                                  |     ||---------------              otherwise             |
      ||/       2/x\\                                   |     ||              2                                    |
      |||1 + cot |-||                                   |     || /       2/x\\                                     |
      ||\        \2//                                   |     || |1 + cot |-||                                     |
      \\                                                /     \\ \        \2//                                     /

$$\left(- 8 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - \left(8 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + 5$$

      //      0         for And(im(x) = 0, x mod pi = 0)\     //      1         for And(im(x) = 0, x mod 2*pi = 0)\
      ||                                                |     ||                                                  |
      ||       2/x\                                     |     ||             2                                    |
      ||  4*tan |-|                                     |     ||/       2/x\\                                     |
      ||        \2/                                     |     |||1 - tan |-||                                     |
5 - 8*|<--------------             otherwise            | - 8*|<\        \2//                                     |
      ||             2                                  |     ||--------------              otherwise             |
      ||/       2/x\\                                   |     ||             2                                    |
      |||1 + tan |-||                                   |     ||/       2/x\\                                     |
      ||\        \2//                                   |     |||1 + tan |-||                                     |
      \\                                                /     \\\        \2//                                     /

$$\left(- 8 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - \left(8 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + 5$$

      //     0        for And(im(x) = 0, x mod pi = 0)\     //   1     for And(im(x) = 0, x mod 2*pi = 0)\
      ||                                              |     ||                                           |
5 - 8*|<   2/    pi\                                  | - 8*|<   2                                       |
      ||cos |x - --|             otherwise            |     ||cos (x)              otherwise             |
      \\    \    2 /                                  /     \\                                           /

$$\left(- 8 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\cos^{2}{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) - \left(8 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + 5$$

Комбинаторика

         2           2   
5 - 8*cos (x) - 8*sin (x)

$$- 8 \sin^{2}{\left (x \right )} - 8 \cos^{2}{\left (x \right )} + 5$$

Раскрыть выражение

         2           2   
5 - 8*cos (x) - 8*sin (x)

$$- 8 \sin^{2}{\left (x \right )} - 8 \cos^{2}{\left (x \right )} + 5$$