25*cos(x)^2*sin(y)^2+25*cos(x)^2*cos(y)^2еслиy=1 (упростите выражение)

Вы ввели

      2       2            2       2   
25*cos (x)*sin (y) + 25*cos (x)*cos (y)

$$25 \sin^{2}{\left(y \right)} \cos^{2}{\left(x \right)} + 25 \cos^{2}{\left(x \right)} \cos^{2}{\left(y \right)}$$

Подстановка условия

25*cos(x)^2*sin(y)^2 + 25*cos(x)^2*cos(y)^2 при y = 1

подставляем

      2       2            2       2   
25*cos (x)*sin (y) + 25*cos (x)*cos (y)

$$25 \sin^{2}{\left(y \right)} \cos^{2}{\left(x \right)} + 25 \cos^{2}{\left(x \right)} \cos^{2}{\left(y \right)}$$

      2   
25*cos (x)

$$25 \cos^{2}{\left(x \right)}$$

переменные

y = 1

$$y = 1$$

      2   
25*cos (x)

$$25 \cos^{2}{\left(x \right)}$$

Степени

                                                      2                  
                                        / I*x    -I*x\                  2
                 2               2      |e      e    |  /   -I*y    I*y\ 
   / I*x    -I*x\  / I*y    -I*y\    25*|---- + -----| *\- e     + e   / 
   |e      e    |  |e      e    |       \ 2       2  /                   
25*|---- + -----| *|---- + -----|  - ------------------------------------
   \ 2       2  /  \ 2       2  /                     4                  

$$25 \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2} \left(\frac{e^{i y}}{2} + \frac{e^{- i y}}{2}\right)^{2} - \frac{25 \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2} \left(e^{i y} - e^{- i y}\right)^{2}}{4}$$

Численный ответ

25.0*cos(x)^2*cos(y)^2 + 25.0*cos(x)^2*sin(y)^2

Рациональный знаменатель

      2       2            2       2   
25*cos (x)*cos (y) + 25*cos (x)*sin (y)

$$25 \sin^{2}{\left (y \right )} \cos^{2}{\left (x \right )} + 25 \cos^{2}{\left (x \right )} \cos^{2}{\left (y \right )}$$

Объединение рациональных выражений

      2    /   2         2   \
25*cos (x)*\cos (y) + sin (y)/

$$25 \left(\sin^{2}{\left(y \right)} + \cos^{2}{\left(y \right)}\right) \cos^{2}{\left(x \right)}$$

Общее упрощение

      2   
25*cos (x)

$$25 \cos^{2}{\left(x \right)}$$

Собрать выражение

25   25*cos(2*x)
-- + -----------
2         2     

$$\frac{25}{2} \cos{\left (2 x \right )} + \frac{25}{2}$$

Комбинаторика

      2    /   2         2   \
25*cos (x)*\cos (y) + sin (y)/

$$25 \left(\sin^{2}{\left(y \right)} + \cos^{2}{\left(y \right)}\right) \cos^{2}{\left(x \right)}$$

Общий знаменатель

      2       2            2       2   
25*cos (x)*cos (y) + 25*cos (x)*sin (y)

$$25 \sin^{2}{\left (y \right )} \cos^{2}{\left (x \right )} + 25 \cos^{2}{\left (x \right )} \cos^{2}{\left (y \right )}$$

Тригонометрическая часть

       25                  25         
--------------- + --------------------
   2       2         2       2/    pi\
sec (x)*sec (y)   sec (x)*sec |y - --|
                              \    2 /

$$\frac{25}{\sec^{2}{\left(x \right)} \sec^{2}{\left(y - \frac{\pi}{2} \right)}} + \frac{25}{\sec^{2}{\left(x \right)} \sec^{2}{\left(y \right)}}$$

   //   0     for And(im(y) = 0, y mod pi = 0)\ //     1        for And(im(x) = 0, x mod 2*pi = 0)\      //     1        for And(im(x) = 0, x mod 2*pi = 0)\ //     1        for And(im(y) = 0, y mod 2*pi = 0)\
   ||                                         | ||                                                |      ||                                                | ||                                                |
   ||   1                                     | ||     1                                          |      ||     1                                          | ||     1                                          |
25*|<-------             otherwise            |*|<------------              otherwise             | + 25*|<------------              otherwise             |*|<------------              otherwise             |
   ||   2                                     | ||   2/pi    \                                    |      ||   2/pi    \                                    | ||   2/pi    \                                    |
   ||csc (y)                                  | ||csc |-- - x|                                    |      ||csc |-- - x|                                    | ||csc |-- - y|                                    |
   \\                                         / \\    \2     /                                    /      \\    \2     /                                    / \\    \2     /                                    /

$$\left(25 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(y \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- y + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)$$

25   25*cos(2*x)
-- + -----------
2         2     

$$\frac{25 \cos{\left(2 x \right)}}{2} + \frac{25}{2}$$

                2              2                      2         
   /       2/x\\  /       2/y\\          /       2/x\\     2/y\ 
25*|1 - tan |-|| *|1 - tan |-||      100*|1 - tan |-|| *tan |-| 
   \        \2//  \        \2//          \        \2//      \2/ 
-------------------------------- + -----------------------------
              2              2                  2              2
 /       2/x\\  /       2/y\\      /       2/x\\  /       2/y\\ 
 |1 + tan |-|| *|1 + tan |-||      |1 + tan |-|| *|1 + tan |-|| 
 \        \2//  \        \2//      \        \2//  \        \2// 

$$\frac{25 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2} \left(1 - \tan^{2}{\left(\frac{y}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{y}{2} \right)} + 1\right)^{2}} + \frac{100 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2} \tan^{2}{\left(\frac{y}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{y}{2} \right)} + 1\right)^{2}}$$

                    2                   2
   /          /x\  \  /           /x\  \ 
   |       sec|-|  |  |        sec|-|  | 
   |          \2/  |  |           \2/  | 
25*|1 + -----------| *|-1 + -----------| 
   |       /x   pi\|  |        /x   pi\| 
   |    sec|- - --||  |     sec|- - --|| 
   \       \2   2 //  \        \2   2 // 
-----------------------------------------
                    4/x\                 
                 sec |-|                 
                     \2/                 

$$\frac{25 \left(\frac{\sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2} \left(\frac{\sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}}{\sec^{4}{\left(\frac{x}{2} \right)}}$$

   //   0     for And(im(y) = 0, y mod pi = 0)\ //     1        for And(im(x) = 0, x mod 2*pi = 0)\      //     1        for And(im(x) = 0, x mod 2*pi = 0)\ //     1        for And(im(y) = 0, y mod 2*pi = 0)\
   ||                                         | ||                                                |      ||                                                | ||                                                |
25*|<   2                                     |*|<   2/    pi\                                    | + 25*|<   2/    pi\                                    |*|<   2/    pi\                                    |
   ||sin (y)             otherwise            | ||sin |x + --|              otherwise             |      ||sin |x + --|              otherwise             | ||sin |y + --|              otherwise             |
   \\                                         / \\    \    2 /                                    /      \\    \    2 /                                    / \\    \    2 /                                    /

$$\left(25 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod \pi = 0 \\\sin^{2}{\left(y \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\sin^{2}{\left(x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\sin^{2}{\left(x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod 2 \pi = 0 \\\sin^{2}{\left(y + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right)$$

                2
   /       2/x\\ 
25*|1 - tan |-|| 
   \        \2// 
-----------------
               2 
  /       2/x\\  
  |1 + tan |-||  
  \        \2//  

$$\frac{25 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$

      2       2            2       2/    pi\
25*cos (x)*cos (y) + 25*cos (x)*cos |y - --|
                                    \    2 /

$$25 \cos^{2}{\left(x \right)} \cos^{2}{\left(y \right)} + 25 \cos^{2}{\left(x \right)} \cos^{2}{\left(y - \frac{\pi}{2} \right)}$$

       25                25      
--------------- + ---------------
   2       2         2       2   
csc (y)*sec (x)   sec (x)*sec (y)

$$\frac{25}{\sec^{2}{\left(x \right)} \sec^{2}{\left(y \right)}} + \frac{25}{\csc^{2}{\left(y \right)} \sec^{2}{\left(x \right)}}$$

      2   
25*cos (x)

$$25 \cos^{2}{\left(x \right)}$$

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

$$25 \sin^{2}{\left(x + \frac{\pi}{2} \right)}$$

                                //                                /           x             \\
                                ||           1             for And|im(x) = 0, - mod 2*pi = 0||
               2              2 ||                                \           2             /|
   /       /x\\  /        /x\\  ||                                                           |
25*|1 + tan|-|| *|-1 + tan|-|| *|<              4                                            |
   \       \2//  \        \2//  ||/        2/x\\     8/x\                                    |
                                |||-1 + cot |-|| *sin |-|              otherwise             |
                                ||\         \4//      \4/                                    |
                                \\                                                           /

$$25 \left(\tan{\left(\frac{x}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right)$$

                    2                   2        
   /       /x   pi\\  /        /x   pi\\         
   |    cos|- - --||  |     cos|- - --||         
   |       \2   2 /|  |        \2   2 /|     4/x\
25*|1 + -----------| *|-1 + -----------| *cos |-|
   |          /x\  |  |           /x\  |      \2/
   |       cos|-|  |  |        cos|-|  |         
   \          \2/  /  \           \2/  /         

$$25 \left(-1 + \frac{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}}\right)^{2} \left(1 + \frac{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}}\right)^{2} \cos^{4}{\left(\frac{x}{2} \right)}$$

                        2                       2        
   /      1        1   \  /       1        1   \     4/x\
25*|1 + ------ - ------| *|-1 + ------ - ------| *cos |-|
   \    sin(x)   tan(x)/  \     sin(x)   tan(x)/      \2/

$$25 \left(-1 - \frac{1}{\tan{\left(x \right)}} + \frac{1}{\sin{\left(x \right)}}\right)^{2} \left(1 - \frac{1}{\tan{\left(x \right)}} + \frac{1}{\sin{\left(x \right)}}\right)^{2} \cos^{4}{\left(\frac{x}{2} \right)}$$

      2       2/    pi\         2/    pi\    2/    pi\
25*sin (y)*sin |x + --| + 25*sin |x + --|*sin |y + --|
               \    2 /          \    2 /     \    2 /

$$25 \sin^{2}{\left(y \right)} \sin^{2}{\left(x + \frac{\pi}{2} \right)} + 25 \sin^{2}{\left(x + \frac{\pi}{2} \right)} \sin^{2}{\left(y + \frac{\pi}{2} \right)}$$

                    2                   2
   /       /pi   x\\  /        /pi   x\\ 
   |    csc|-- - -||  |     csc|-- - -|| 
   |       \2    2/|  |        \2    2/| 
25*|1 + -----------| *|-1 + -----------| 
   |          /x\  |  |           /x\  | 
   |       csc|-|  |  |        csc|-|  | 
   \          \2/  /  \           \2/  / 
-----------------------------------------
                  4/pi   x\              
               csc |-- - -|              
                   \2    2/              

$$\frac{25 \left(-1 + \frac{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}}\right)^{2} \left(1 + \frac{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}}\right)^{2}}{\csc^{4}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}$$

   //   1     for And(im(x) = 0, x mod 2*pi = 0)\
   ||                                           |
25*|<   2                                       |
   ||cos (x)              otherwise             |
   \\                                           /

$$25 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)$$

   //      0         for And(im(y) = 0, y mod pi = 0)\ //       1         for And(im(x) = 0, x mod 2*pi = 0)\      //       1         for And(im(x) = 0, x mod 2*pi = 0)\ //       1         for And(im(y) = 0, y mod 2*pi = 0)\
   ||                                                | ||                                                   |      ||                                                   | ||                                                   |
   ||       2/y\                                     | ||              2                                    |      ||              2                                    | ||              2                                    |
   ||  4*cot |-|                                     | ||/        2/x\\                                     |      ||/        2/x\\                                     | ||/        2/y\\                                     |
   ||        \2/                                     | |||-1 + cot |-||                                     |      |||-1 + cot |-||                                     | |||-1 + cot |-||                                     |
25*|<--------------             otherwise            |*|<\         \2//                                     | + 25*|<\         \2//                                     |*|<\         \2//                                     |
   ||             2                                  | ||---------------              otherwise             |      ||---------------              otherwise             | ||---------------              otherwise             |
   ||/       2/y\\                                   | ||              2                                    |      ||              2                                    | ||              2                                    |
   |||1 + cot |-||                                   | || /       2/x\\                                     |      || /       2/x\\                                     | || /       2/y\\                                     |
   ||\        \2//                                   | || |1 + cot |-||                                     |      || |1 + cot |-||                                     | || |1 + cot |-||                                     |
   \\                                                / \\ \        \2//                                     /      \\ \        \2//                                     / \\ \        \2//                                     /

$$\left(25 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{y}{2} \right)}}{\left(\cot^{2}{\left(\frac{y}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{y}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{y}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$

         25                        25           
-------------------- + -------------------------
   2       2/pi    \      2/pi    \    2/pi    \
csc (y)*csc |-- - x|   csc |-- - x|*csc |-- - y|
            \2     /       \2     /     \2     /

$$\frac{25}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)} \csc^{2}{\left(- y + \frac{\pi}{2} \right)}} + \frac{25}{\csc^{2}{\left(y \right)} \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}$$

   25  
-------
   2   
sec (x)

$$\frac{25}{\sec^{2}{\left(x \right)}}$$

   //       1         for And(im(x) = 0, x mod 2*pi = 0)\
   ||                                                   |
   ||              2                                    |
   ||/        2/x\\                                     |
   |||-1 + cot |-||                                     |
25*|<\         \2//                                     |
   ||---------------              otherwise             |
   ||              2                                    |
   || /       2/x\\                                     |
   || |1 + cot |-||                                     |
   \\ \        \2//                                     /

$$25 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$

     25     
------------
   2/pi    \
csc |-- - x|
    \2     /

$$\frac{25}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}$$

   //     0        for And(im(y) = 0, y mod pi = 0)\ //   1     for And(im(x) = 0, x mod 2*pi = 0)\      //   1     for And(im(x) = 0, x mod 2*pi = 0)\ //   1     for And(im(y) = 0, y mod 2*pi = 0)\
   ||                                              | ||                                           |      ||                                           | ||                                           |
25*|<   2/    pi\                                  |*|<   2                                       | + 25*|<   2                                       |*|<   2                                       |
   ||cos |y - --|             otherwise            | ||cos (x)              otherwise             |      ||cos (x)              otherwise             | ||cos (y)              otherwise             |
   \\    \    2 /                                  / \\                                           /      \\                                           / \\                                           /

$$\left(25 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod \pi = 0 \\\cos^{2}{\left(y - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod 2 \pi = 0 \\\cos^{2}{\left(y \right)} & \text{otherwise} \end{cases}\right)\right)$$

                  2                 2             
   /         2/x\\  /          2/x\\              
   |    2*sin |-||  |     2*sin |-||              
   |          \2/|  |           \2/|     4/pi   x\
25*|1 + ---------| *|-1 + ---------| *sin |-- + -|
   \      sin(x) /  \       sin(x) /      \2    2/

$$25 \left(\frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} - 1\right)^{2} \left(\frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} + 1\right)^{2} \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}$$

   //     0        for And(im(y) = 0, y mod pi = 0)\ //   1     for And(im(x) = 0, x mod 2*pi = 0)\      //   1     for And(im(x) = 0, x mod 2*pi = 0)\ //   1     for And(im(y) = 0, y mod 2*pi = 0)\
   ||                                              | ||                                           |      ||                                           | ||                                           |
   ||     1                                        | ||   1                                       |      ||   1                                       | ||   1                                       |
25*|<------------             otherwise            |*|<-------              otherwise             | + 25*|<-------              otherwise             |*|<-------              otherwise             |
   ||   2/    pi\                                  | ||   2                                       |      ||   2                                       | ||   2                                       |
   ||sec |y - --|                                  | ||sec (x)                                    |      ||sec (x)                                    | ||sec (y)                                    |
   \\    \    2 /                                  / \\                                           /      \\                                           / \\                                           /

$$\left(25 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(y - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(y \right)}} & \text{otherwise} \end{cases}\right)\right)$$

   //      0         for And(im(y) = 0, y mod pi = 0)\ //      1         for And(im(x) = 0, x mod 2*pi = 0)\      //      1         for And(im(x) = 0, x mod 2*pi = 0)\ //      1         for And(im(y) = 0, y mod 2*pi = 0)\
   ||                                                | ||                                                  |      ||                                                  | ||                                                  |
   ||       2/y\                                     | ||             2                                    |      ||             2                                    | ||             2                                    |
   ||  4*tan |-|                                     | ||/       2/x\\                                     |      ||/       2/x\\                                     | ||/       2/y\\                                     |
   ||        \2/                                     | |||1 - tan |-||                                     |      |||1 - tan |-||                                     | |||1 - tan |-||                                     |
25*|<--------------             otherwise            |*|<\        \2//                                     | + 25*|<\        \2//                                     |*|<\        \2//                                     |
   ||             2                                  | ||--------------              otherwise             |      ||--------------              otherwise             | ||--------------              otherwise             |
   ||/       2/y\\                                   | ||             2                                    |      ||             2                                    | ||             2                                    |
   |||1 + tan |-||                                   | ||/       2/x\\                                     |      ||/       2/x\\                                     | ||/       2/y\\                                     |
   ||\        \2//                                   | |||1 + tan |-||                                     |      |||1 + tan |-||                                     | |||1 + tan |-||                                     |
   \\                                                / \\\        \2//                                     /      \\\        \2//                                     / \\\        \2//                                     /

$$\left(25 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{y}{2} \right)}}{\left(\tan^{2}{\left(\frac{y}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod 2 \pi = 0 \\\frac{\left(1 - \tan^{2}{\left(\frac{y}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{y}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$

                4             2              2        
   /       2/x\\  /       /x\\  /        /x\\     8/x\
25*|1 - tan |-|| *|1 + tan|-|| *|-1 + tan|-|| *cos |-|
   \        \4//  \       \2//  \        \2//      \4/

$$25 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{4} \left(\tan{\left(\frac{x}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \cos^{8}{\left(\frac{x}{4} \right)}$$

                                //                        /           x             \\
                                ||       1         for And|im(x) = 0, - mod 2*pi = 0||
                                ||                        \           2             /|
                                ||                                                   |
               2              2 ||              4                                    |
   /      1   \  /       1   \  ||/        2/x\\                                     |
25*|1 + ------| *|-1 + ------| *|<|-1 + cot |-||                                     |
   |       /x\|  |        /x\|  ||\         \4//                                     |
   |    cot|-||  |     cot|-||  ||---------------              otherwise             |
   \       \2//  \        \2//  ||              4                                    |
                                || /       2/x\\                                     |
                                || |1 + cot |-||                                     |
                                \\ \        \4//                                     /

$$25 \left(-1 + \frac{1}{\cot{\left(\frac{x}{2} \right)}}\right)^{2} \left(1 + \frac{1}{\cot{\left(\frac{x}{2} \right)}}\right)^{2} \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{4}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)$$

   //                        0                          for And(im(y) = 0, y mod pi = 0)\ //                         1                            for And(im(x) = 0, x mod 2*pi = 0)\      //                         1                            for And(im(x) = 0, x mod 2*pi = 0)\ //                         1                            for And(im(y) = 0, y mod 2*pi = 0)\
   ||                                                                                   | ||                                                                                        |      ||                                                                                        | ||                                                                                        |
   ||/      0         for And(im(y) = 0, y mod pi = 0)                                  | ||/       1         for And(im(x) = 0, x mod 2*pi = 0)                                    |      ||/       1         for And(im(x) = 0, x mod 2*pi = 0)                                    | ||/       1         for And(im(y) = 0, y mod 2*pi = 0)                                    |
   |||                                                                                  | |||                                                                                       |      |||                                                                                       | |||                                                                                       |
   |||       2/y\                                                                       | |||              2                                                                        |      |||              2                                                                        | |||              2                                                                        |
   |||  4*cot |-|                                                                       | |||/        2/x\\                                                                         |      |||/        2/x\\                                                                         | |||/        2/y\\                                                                         |
25*|<|        \2/                                                                       |*|<||-1 + cot |-||                                                                         | + 25*|<||-1 + cot |-||                                                                         |*|<||-1 + cot |-||                                                                         |
   ||<--------------             otherwise                         otherwise            | ||<\         \2//                                                   otherwise             |      ||<\         \2//                                                   otherwise             | ||<\         \2//                                                   otherwise             |
   |||             2                                                                    | |||---------------              otherwise                                                 |      |||---------------              otherwise                                                 | |||---------------              otherwise                                                 |
   |||/       2/y\\                                                                     | |||              2                                                                        |      |||              2                                                                        | |||              2                                                                        |
   ||||1 + cot |-||                                                                     | ||| /       2/x\\                                                                         |      ||| /       2/x\\                                                                         | ||| /       2/y\\                                                                         |
   |||\        \2//                                                                     | ||| |1 + cot |-||                                                                         |      ||| |1 + cot |-||                                                                         | ||| |1 + cot |-||                                                                         |
   \\\                                                                                  / \\\ \        \2//                                                                         /      \\\ \        \2//                                                                         / \\\ \        \2//                                                                         /

$$\left(25 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{y}{2} \right)}}{\left(\cot^{2}{\left(\frac{y}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{y}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{y}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$

   //                    0                       for And(im(y) = 0, y mod pi = 0)\ //                     1                        for And(im(x) = 0, x mod 2*pi = 0)\      //                     1                        for And(im(x) = 0, x mod 2*pi = 0)\ //                     1                        for And(im(y) = 0, y mod 2*pi = 0)\
   ||                                                                            | ||                                                                                |      ||                                                                                | ||                                                                                |
   ||/   0     for And(im(y) = 0, y mod pi = 0)                                  | ||/   1     for And(im(x) = 0, x mod 2*pi = 0)                                    |      ||/   1     for And(im(x) = 0, x mod 2*pi = 0)                                    | ||/   1     for And(im(y) = 0, y mod 2*pi = 0)                                    |
25*|<|                                                                           |*|<|                                                                               | + 25*|<|                                                                               |*|<|                                                                               |
   ||<   2                                                  otherwise            | ||<   2                                                     otherwise             |      ||<   2                                                     otherwise             | ||<   2                                                     otherwise             |
   |||sin (y)             otherwise                                              | |||cos (x)              otherwise                                                 |      |||cos (x)              otherwise                                                 | |||cos (y)              otherwise                                                 |
   \\\                                                                           / \\\                                                                               /      \\\                                                                               / \\\                                                                               /

$$\left(25 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod \pi = 0 \\\sin^{2}{\left(y \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod 2 \pi = 0 \\\cos^{2}{\left(y \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$

                4             2              2
   /       2/x\\  /       /x\\  /        /x\\ 
25*|1 - tan |-|| *|1 + tan|-|| *|-1 + tan|-|| 
   \        \4//  \       \2//  \        \2// 
----------------------------------------------
                             4                
                /       2/x\\                 
                |1 + tan |-||                 
                \        \4//                 

$$\frac{25 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{4} \left(\tan{\left(\frac{x}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4}}$$

   //   0     for And(im(y) = 0, y mod pi = 0)\ //   1     for And(im(x) = 0, x mod 2*pi = 0)\      //   1     for And(im(x) = 0, x mod 2*pi = 0)\ //   1     for And(im(y) = 0, y mod 2*pi = 0)\
   ||                                         | ||                                           |      ||                                           | ||                                           |
25*|<   2                                     |*|<   2                                       | + 25*|<   2                                       |*|<   2                                       |
   ||sin (y)             otherwise            | ||cos (x)              otherwise             |      ||cos (x)              otherwise             | ||cos (y)              otherwise             |
   \\                                         / \\                                           /      \\                                           / \\                                           /

$$\left(25 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod \pi = 0 \\\sin^{2}{\left(y \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod 2 \pi = 0 \\\cos^{2}{\left(y \right)} & \text{otherwise} \end{cases}\right)\right)$$

               2              2        
   /       /x\\  /        /x\\     4/x\
25*|1 + tan|-|| *|-1 + tan|-|| *cos |-|
   \       \2//  \        \2//      \2/

$$25 \left(\tan{\left(\frac{x}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \cos^{4}{\left(\frac{x}{2} \right)}$$

Раскрыть выражение

      2       2            2       2   
25*cos (x)*cos (y) + 25*cos (x)*sin (y)

$$25 \sin^{2}{\left (y \right )} \cos^{2}{\left (x \right )} + 25 \cos^{2}{\left (x \right )} \cos^{2}{\left (y \right )}$$