1-sin(x)^2-(-1)*(2*cos(x)^2)еслиx=3 (упростите выражение)

Вы ввели

       2      -       2   
1 - sin (x) - 1*2*cos (x) 

$$- \sin^{2}{\left(x \right)} - \left(-1\right) 2 \cos^{2}{\left(x \right)} + 1$$

Подстановка условия

1 - sin(x)^2 - (-1)*2*cos(x)^2 при x = 3

подставляем

       2      -       2   
1 - sin (x) - 1*2*cos (x) 

$$- \sin^{2}{\left(x \right)} - \left(-1\right) 2 \cos^{2}{\left(x \right)} + 1$$

     2   
3*cos (x)

$$3 \cos^{2}{\left(x \right)}$$

переменные

x = 3

$$x = 3$$

     2     
3*cos ((3))

$$3 \cos^{2}{\left((3) \right)}$$

     2   
3*cos (3)

$$3 \cos^{2}{\left(3 \right)}$$

Степени

                    2                   2
      / I*x    -I*x\    /   -I*x    I*x\ 
      |e      e    |    \- e     + e   / 
1 + 2*|---- + -----|  + -----------------
      \ 2       2  /            4        

$$2 \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2} + \frac{\left(e^{i x} - e^{- i x}\right)^{2}}{4} + 1$$

       2           2   
1 - sin (x) + 2*cos (x)

$$- \sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)} + 1$$

Численный ответ

1.0 - sin(x)^2 + 2.0*cos(x)^2

Рациональный знаменатель

       2           2   
1 - sin (x) + 2*cos (x)

$$- \sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)} + 1$$

Объединение рациональных выражений

       2           2   
1 - sin (x) + 2*cos (x)

$$- \sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)} + 1$$

Общее упрощение

     2   
3*cos (x)

$$3 \cos^{2}{\left(x \right)}$$

Собрать выражение

       2           2   
1 - sin (x) + 2*cos (x)

$$- \sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)} + 1$$

Комбинаторика

       2           2   
1 - sin (x) + 2*cos (x)

$$- \sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)} + 1$$

Общий знаменатель

       2           2   
1 - sin (x) + 2*cos (x)

$$- \sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)} + 1$$

Тригонометрическая часть

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

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

    //      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 |-||                                     |
1 - |<--------------             otherwise            | + 2*|<\        \2//                                     |
    ||             2                                  |     ||--------------              otherwise             |
    ||/       2/x\\                                   |     ||             2                                    |
    |||1 + tan |-||                                   |     ||/       2/x\\                                     |
    ||\        \2//                                   |     |||1 + tan |-||                                     |
    \\                                                /     \\\        \2//                                     /

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

                                    2
           2/x\        /       2/x\\ 
      4*tan |-|      2*|1 - tan |-|| 
            \2/        \        \2// 
1 - -------------- + ----------------
                 2                 2 
    /       2/x\\     /       2/x\\  
    |1 + tan |-||     |1 + tan |-||  
    \        \2//     \        \2//  

$$\frac{2 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 1 - \frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$

3   3*cos(2*x)
- + ----------
2       2     

$$\frac{3 \cos{\left(2 x \right)}}{2} + \frac{3}{2}$$

       2           2   
1 - sin (x) + 2*cos (x)

$$- \sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)} + 1$$

   3   
-------
   2   
sec (x)

$$\frac{3}{\sec^{2}{\left(x \right)}}$$

  //   1     for And(im(x) = 0, x mod 2*pi = 0)\
  ||                                           |
3*|<   2                                       |
  ||cos (x)              otherwise             |
  \\                                           /

$$3 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)$$

     2/    pi\
3*sin |x + --|
      \    2 /

$$3 \sin^{2}{\left(x + \frac{\pi}{2} \right)}$$

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

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

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

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

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

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

    //                    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)                                    |
1 - |<|                                                                           | + 2*|<|                                                                               |
    ||<   2                                                  otherwise            |     ||<   2                                                     otherwise             |
    |||sin (x)             otherwise                                              |     |||cos (x)              otherwise                                                 |
    \\\                                                                           /     \\\                                                                               /

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

         1            2   
1 - ------------ + -------
       2/    pi\      2   
    sec |x - --|   sec (x)
        \    2 /          

$$1 - \frac{1}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + \frac{2}{\sec^{2}{\left(x \right)}}$$

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

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

       1         2   
1 - ------- + -------
       2         2   
    csc (x)   sec (x)

$$1 + \frac{2}{\sec^{2}{\left(x \right)}} - \frac{1}{\csc^{2}{\left(x \right)}}$$

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

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

       1           2      
1 - ------- + ------------
       2         2/pi    \
    csc (x)   csc |-- - x|
                  \2     /

$$1 + \frac{2}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} - \frac{1}{\csc^{2}{\left(x \right)}}$$

       2/    pi\        2   
1 - cos |x - --| + 2*cos (x)
        \    2 /            

$$2 \cos^{2}{\left(x \right)} - \cos^{2}{\left(x - \frac{\pi}{2} \right)} + 1$$

               2
  /       2/x\\ 
3*|1 - tan |-|| 
  \        \2// 
----------------
              2 
 /       2/x\\  
 |1 + tan |-||  
 \        \2//  

$$\frac{3 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$

     2   
3*cos (x)

$$3 \cos^{2}{\left(x \right)}$$

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

$$- \sin^{2}{\left(x \right)} + 2 \sin^{2}{\left(x + \frac{\pi}{2} \right)} + 1$$

     3      
------------
   2/pi    \
csc |-- - x|
    \2     /

$$\frac{3}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}$$

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

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

  //       1         for And(im(x) = 0, x mod 2*pi = 0)\
  ||                                                   |
  ||              2                                    |
  ||/        2/x\\                                     |
  |||-1 + cot |-||                                     |
3*|<\         \2//                                     |
  ||---------------              otherwise             |
  ||              2                                    |
  || /       2/x\\                                     |
  || |1 + cot |-||                                     |
  \\ \        \2//                                     /

$$3 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$

Раскрыть выражение

       2           2   
1 - sin (x) + 2*cos (x)

$$- \sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)} + 1$$