Найти значение выражения cos(z+y)-sin(z)*cos(y)еслиy=3/2 (косинус от (z плюс у) минус синус от (z) умножить на косинус от (у)если у равно 3 делить на 2) [Есть ответ!]

cos(z+y)-sin(z)*cos(y)еслиy=3/2 (упростите выражение)

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Решение

Вы ввели [src]
cos(z + y) - sin(z)*cos(y)
$$- \sin{\left(z \right)} \cos{\left(y \right)} + \cos{\left(y + z \right)}$$
Подстановка условия [src]
cos(z + y) - sin(z)*cos(y) при y = 3/2
подставляем
cos(z + y) - sin(z)*cos(y)
$$- \sin{\left(z \right)} \cos{\left(y \right)} + \cos{\left(y + z \right)}$$
-cos(y)*sin(z) + cos(y + z)
$$- \sin{\left(z \right)} \cos{\left(y \right)} + \cos{\left(y + z \right)}$$
переменные
y = 3/2
$$y = \frac{3}{2}$$
-cos((3/2))*sin(z) + cos((3/2) + z)
$$- \sin{\left(z \right)} \cos{\left((3/2) \right)} + \cos{\left((3/2) + z \right)}$$
-cos(3/2)*sin(z) + cos(3/2 + z)
$$- \sin{\left(z \right)} \cos{\left(\frac{3}{2} \right)} + \cos{\left(z + \frac{3}{2} \right)}$$
Степени [src]
                             / I*y    -I*y\                 
                             |e      e    | /   -I*z    I*z\
 I*(y + z)    I*(-y - z)   I*|---- + -----|*\- e     + e   /
e            e               \ 2       2  /                 
---------- + ----------- + ---------------------------------
    2             2                        2                
$$\frac{i \left(\frac{e^{i y}}{2} + \frac{e^{- i y}}{2}\right) \left(e^{i z} - e^{- i z}\right)}{2} + \frac{e^{i \left(- y - z\right)}}{2} + \frac{e^{i \left(y + z\right)}}{2}$$
Численный ответ [src]
-cos(y)*sin(z) + cos(z + y)
Собрать выражение [src]
sin(y - z)   sin(y + z)             
---------- - ---------- + cos(y + z)
    2            2                  
$$\frac{1}{2} \sin{\left (y - z \right )} - \frac{1}{2} \sin{\left (y + z \right )} + \cos{\left (y + z \right )}$$
Тригонометрическая часть [src]
            /    pi\             
- cos(y)*cos|z - --| + cos(y + z)
            \    2 /             
$$- \cos{\left(y \right)} \cos{\left(z - \frac{\pi}{2} \right)} + \cos{\left(y + z \right)}$$
  //     0       for And(im(z) = 0, z mod pi = 0)\                                                                                                                
  ||                                             | //  1     for And(im(y) = 0, y mod 2*pi = 0)\   //    1       for And(im(y) + im(z) = 0, (y + z) mod 2*pi = 0)\
- |<   /    pi\                                  |*|<                                          | + |<                                                            |
  ||cos|z - --|             otherwise            | \\cos(y)              otherwise             /   \\cos(y + z)                     otherwise                    /
  \\   \    2 /                                  /                                                                                                                
$$\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(z\right)} = 0 \wedge z \bmod \pi = 0 \\\cos{\left(z - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod 2 \pi = 0 \\\cos{\left(y \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} + \operatorname{im}{\left(z\right)} = 0 \wedge \left(y + z\right) \bmod 2 \pi = 0 \\\cos{\left(y + z \right)} & \text{otherwise} \end{cases}\right)$$
       2/y   z\        /       2/y\\    /z\  
1 - tan |- + -|      2*|1 - tan |-||*tan|-|  
        \2   2/        \        \2//    \2/  
--------------- - ---------------------------
       2/y   z\   /       2/y\\ /       2/z\\
1 + tan |- + -|   |1 + tan |-||*|1 + tan |-||
        \2   2/   \        \2// \        \2//
$$- \frac{2 \cdot \left(1 - \tan^{2}{\left(\frac{y}{2} \right)}\right) \tan{\left(\frac{z}{2} \right)}}{\left(\tan^{2}{\left(\frac{y}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{z}{2} \right)} + 1\right)} + \frac{1 - \tan^{2}{\left(\frac{y}{2} + \frac{z}{2} \right)}}{\tan^{2}{\left(\frac{y}{2} + \frac{z}{2} \right)} + 1}$$
  //                      0                         for And(im(z) = 0, z mod pi = 0)\ //                        1                          for And(im(y) = 0, y mod 2*pi = 0)\   //                                 1                                   for And(im(y) + im(z) = 0, (y + z) mod 2*pi = 0)\
  ||                                                                                | ||                                                                                     |   ||                                                                                                                     |
  ||/     0       for And(im(z) = 0, z mod pi = 0)                                  | ||/     1        for And(im(y) = 0, y mod 2*pi = 0)                                    |   ||/       1          for And(im(y) + im(z) = 0, (y + z) mod 2*pi = 0)                                                  |
  |||                                                                               | |||                                                                                    |   |||                                                                                                                    |
  |||       /z\                                                                     | |||        2/y\                                                                        |   |||        2/y   z\                                                                                                    |
- |<|  2*cot|-|                                                                     |*|<|-1 + cot |-|                                                                        | + |<|-1 + cot |- + -|                                                                                                    |
  ||<       \2/                                                otherwise            | ||<         \2/                                                  otherwise             |   ||<         \2   2/                                                                       otherwise                    |
  |||-----------             otherwise                                              | |||------------              otherwise                                                 |   |||----------------                     otherwise                                                                      |
  |||       2/z\                                                                    | |||       2/y\                                                                         |   |||       2/y   z\                                                                                                     |
  |||1 + cot |-|                                                                    | |||1 + cot |-|                                                                         |   |||1 + cot |- + -|                                                                                                     |
  \\\        \2/                                                                    / \\\        \2/                                                                         /   \\\        \2   2/                                                                                                     /
$$\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(z\right)} = 0 \wedge z \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(z\right)} = 0 \wedge z \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{z}{2} \right)}}{\cot^{2}{\left(\frac{z}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{y}{2} \right)} - 1}{\cot^{2}{\left(\frac{y}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} + \operatorname{im}{\left(z\right)} = 0 \wedge \left(y + z\right) \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} + \operatorname{im}{\left(z\right)} = 0 \wedge \left(y + z\right) \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{y}{2} + \frac{z}{2} \right)} - 1}{\cot^{2}{\left(\frac{y}{2} + \frac{z}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
  //                    0                      for And(im(z) = 0, z mod pi = 0)\ //                     1                       for And(im(y) = 0, y mod 2*pi = 0)\   //                              1                                for And(im(y) + im(z) = 0, (y + z) mod 2*pi = 0)\
  ||                                                                           | ||                                                                               |   ||                                                                                                               |
- |
$$\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(z\right)} = 0 \wedge z \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(z\right)} = 0 \wedge z \bmod \pi = 0 \\\sin{\left(z \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod 2 \pi = 0 \\\cos{\left(y \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} + \operatorname{im}{\left(z\right)} = 0 \wedge \left(y + z\right) \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} + \operatorname{im}{\left(z\right)} = 0 \wedge \left(y + z\right) \bmod 2 \pi = 0 \\\cos{\left(y + z \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
                                    2/y   z\
                             1 - tan |- + -|
  -sin(y - z) + sin(y + z)           \2   2/
- ------------------------ + ---------------
             2                      2/y   z\
                             1 + tan |- + -|
                                     \2   2/
$$\frac{1 - \tan^{2}{\left(\frac{y}{2} + \frac{z}{2} \right)}}{\tan^{2}{\left(\frac{y}{2} + \frac{z}{2} \right)} + 1} - \frac{- \sin{\left(y - z \right)} + \sin{\left(y + z \right)}}{2}$$
  //  0     for And(im(z) = 0, z mod pi = 0)\ //  1     for And(im(y) = 0, y mod 2*pi = 0)\   //    1       for And(im(y) + im(z) = 0, (y + z) mod 2*pi = 0)\
- |<                                        |*|<                                          | + |<                                                            |
  \\sin(z)             otherwise            / \\cos(y)              otherwise             /   \\cos(y + z)                     otherwise                    /
$$\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(z\right)} = 0 \wedge z \bmod \pi = 0 \\\sin{\left(z \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod 2 \pi = 0 \\\cos{\left(y \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} + \operatorname{im}{\left(z\right)} = 0 \wedge \left(y + z\right) \bmod 2 \pi = 0 \\\cos{\left(y + z \right)} & \text{otherwise} \end{cases}\right)$$
                                   4/y   z\
                              4*sin |- + -|
                                    \2   2/
                          1 - -------------
                                  2        
sin(y - z)   sin(y + z)        sin (y + z) 
---------- - ---------- + -----------------
    2            2                 4/y   z\
                              4*sin |- + -|
                                    \2   2/
                          1 + -------------
                                  2        
                               sin (y + z) 
$$\frac{- \frac{4 \sin^{4}{\left(\frac{y}{2} + \frac{z}{2} \right)}}{\sin^{2}{\left(y + z \right)}} + 1}{\frac{4 \sin^{4}{\left(\frac{y}{2} + \frac{z}{2} \right)}}{\sin^{2}{\left(y + z \right)}} + 1} + \frac{\sin{\left(y - z \right)}}{2} - \frac{\sin{\left(y + z \right)}}{2}$$
  //     0       for And(im(z) = 0, z mod pi = 0)\ //     1       for And(im(y) = 0, y mod 2*pi = 0)\   //       1         for And(im(y) + im(z) = 0, (y + z) mod 2*pi = 0)\
  ||                                             | ||                                               |   ||                                                                 |
  ||       /z\                                   | ||       2/y\                                    |   ||       2/y   z\                                                  |
  ||  2*tan|-|                                   | ||1 - tan |-|                                    |   ||1 - tan |- + -|                                                  |
- |<       \2/                                   |*|<        \2/                                    | + |<        \2   2/                                                  |
  ||-----------             otherwise            | ||-----------              otherwise             |   ||---------------                     otherwise                    |
  ||       2/z\                                  | ||       2/y\                                    |   ||       2/y   z\                                                  |
  ||1 + tan |-|                                  | ||1 + tan |-|                                    |   ||1 + tan |- + -|                                                  |
  \\        \2/                                  / \\        \2/                                    /   \\        \2   2/                                                  /
$$\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(z\right)} = 0 \wedge z \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{z}{2} \right)}}{\tan^{2}{\left(\frac{z}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod 2 \pi = 0 \\\frac{1 - \tan^{2}{\left(\frac{y}{2} \right)}}{\tan^{2}{\left(\frac{y}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} + \operatorname{im}{\left(z\right)} = 0 \wedge \left(y + z\right) \bmod 2 \pi = 0 \\\frac{1 - \tan^{2}{\left(\frac{y}{2} + \frac{z}{2} \right)}}{\tan^{2}{\left(\frac{y}{2} + \frac{z}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
                                              //     1       for And(im(y) = 0, y mod 2*pi = 0)\   //       1         for And(im(y) + im(z) = 0, (y + z) mod 2*pi = 0)\
  //  0     for And(im(z) = 0, z mod pi = 0)\ ||                                               |   ||                                                                 |
  ||                                        | ||     1                                         |   ||       1                                                         |
- |<  1                                     |*|<-----------              otherwise             | + |<---------------                     otherwise                    |
  ||------             otherwise            | ||   /pi    \                                    |   ||   /pi        \                                                  |
  \\csc(z)                                  / ||csc|-- - y|                                    |   ||csc|-- - y - z|                                                  |
                                              \\   \2     /                                    /   \\   \2         /                                                  /
$$\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(z\right)} = 0 \wedge z \bmod \pi = 0 \\\frac{1}{\csc{\left(z \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- y + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} + \operatorname{im}{\left(z\right)} = 0 \wedge \left(y + z\right) \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- y - z + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
                                   4/y + z\
                              4*sin |-----|
                                    \  2  /
                          1 - -------------
                                  2        
sin(y - z)   sin(y + z)        sin (y + z) 
---------- - ---------- + -----------------
    2            2                 4/y + z\
                              4*sin |-----|
                                    \  2  /
                          1 + -------------
                                  2        
                               sin (y + z) 
$$\frac{- \frac{4 \sin^{4}{\left(\frac{y + z}{2} \right)}}{\sin^{2}{\left(y + z \right)}} + 1}{\frac{4 \sin^{4}{\left(\frac{y + z}{2} \right)}}{\sin^{2}{\left(y + z \right)}} + 1} + \frac{\sin{\left(y - z \right)}}{2} - \frac{\sin{\left(y + z \right)}}{2}$$
            /    pi\      /        pi\
- sin(z)*sin|y + --| + sin|y + z + --|
            \    2 /      \        2 /
$$- \sin{\left(z \right)} \sin{\left(y + \frac{\pi}{2} \right)} + \sin{\left(y + z + \frac{\pi}{2} \right)}$$
                                              //     1       for And(im(y) = 0, y mod 2*pi = 0)\   //       1         for And(im(y) + im(z) = 0, (y + z) mod 2*pi = 0)\
  //  0     for And(im(z) = 0, z mod pi = 0)\ ||                                               |   ||                                                                 |
- |<                                        |*|<   /    pi\                                    | + |<   /        pi\                                                  |
  \\sin(z)             otherwise            / ||sin|y + --|              otherwise             |   ||sin|y + z + --|                     otherwise                    |
                                              \\   \    2 /                                    /   \\   \        2 /                                                  /
$$\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(z\right)} = 0 \wedge z \bmod \pi = 0 \\\sin{\left(z \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod 2 \pi = 0 \\\sin{\left(y + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} + \operatorname{im}{\left(z\right)} = 0 \wedge \left(y + z\right) \bmod 2 \pi = 0 \\\sin{\left(y + z + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
  //     0       for And(im(z) = 0, z mod pi = 0)\ //     1        for And(im(y) = 0, y mod 2*pi = 0)\   //       1          for And(im(y) + im(z) = 0, (y + z) mod 2*pi = 0)\
  ||                                             | ||                                                |   ||                                                                  |
  ||       /z\                                   | ||        2/y\                                    |   ||        2/y   z\                                                  |
  ||  2*cot|-|                                   | ||-1 + cot |-|                                    |   ||-1 + cot |- + -|                                                  |
- |<       \2/                                   |*|<         \2/                                    | + |<         \2   2/                                                  |
  ||-----------             otherwise            | ||------------              otherwise             |   ||----------------                     otherwise                    |
  ||       2/z\                                  | ||       2/y\                                     |   ||       2/y   z\                                                   |
  ||1 + cot |-|                                  | ||1 + cot |-|                                     |   ||1 + cot |- + -|                                                   |
  \\        \2/                                  / \\        \2/                                     /   \\        \2   2/                                                   /
$$\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(z\right)} = 0 \wedge z \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{z}{2} \right)}}{\cot^{2}{\left(\frac{z}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{y}{2} \right)} - 1}{\cot^{2}{\left(\frac{y}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} + \operatorname{im}{\left(z\right)} = 0 \wedge \left(y + z\right) \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{y}{2} + \frac{z}{2} \right)} - 1}{\cot^{2}{\left(\frac{y}{2} + \frac{z}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
                                                 2/y   z\   
                                              sec |- + -|   
                                                  \2   2/   
                                        1 - ----------------
                                               2/y   z   pi\
                                            sec |- + - - --|
        1                   1                   \2   2   2 /
----------------- - ----------------- + --------------------
     /        pi\        /        pi\            2/y   z\   
2*sec|y - z - --|   2*sec|y + z - --|         sec |- + -|   
     \        2 /        \        2 /             \2   2/   
                                        1 + ----------------
                                               2/y   z   pi\
                                            sec |- + - - --|
                                                \2   2   2 /
$$\frac{- \frac{\sec^{2}{\left(\frac{y}{2} + \frac{z}{2} \right)}}{\sec^{2}{\left(\frac{y}{2} + \frac{z}{2} - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(\frac{y}{2} + \frac{z}{2} \right)}}{\sec^{2}{\left(\frac{y}{2} + \frac{z}{2} - \frac{\pi}{2} \right)}} + 1} - \frac{1}{2 \sec{\left(y + z - \frac{\pi}{2} \right)}} + \frac{1}{2 \sec{\left(y - z - \frac{\pi}{2} \right)}}$$
/    0       for And(-im(z) + im(y) = 0, (y - z) mod pi = 0)   /    0       for And(im(y) + im(z) = 0, (y + z) mod pi = 0)          2/y   z\
<                                                              <                                                             1 - tan |- + -|
\sin(y - z)                     otherwise                      \sin(y + z)                    otherwise                              \2   2/
------------------------------------------------------------ - ----------------------------------------------------------- + ---------------
                             2                                                              2                                       2/y   z\
                                                                                                                             1 + tan |- + -|
                                                                                                                                     \2   2/
$$\frac{1 - \tan^{2}{\left(\frac{y}{2} + \frac{z}{2} \right)}}{\tan^{2}{\left(\frac{y}{2} + \frac{z}{2} \right)} + 1} + \left(\frac{\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(y\right)} - \operatorname{im}{\left(z\right)} = 0 \wedge \left(y - z\right) \bmod \pi = 0 \\\sin{\left(y - z \right)} & \text{otherwise} \end{cases}}{2}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(y\right)} + \operatorname{im}{\left(z\right)} = 0 \wedge \left(y + z\right) \bmod \pi = 0 \\\sin{\left(y + z \right)} & \text{otherwise} \end{cases}}{2}\right)$$
       2/y   z\         /y   z\           /y   z\  
1 - tan |- + -|      tan|- - -|        tan|- + -|  
        \2   2/         \2   2/           \2   2/  
--------------- + --------------- - ---------------
       2/y   z\          2/y   z\          2/y   z\
1 + tan |- + -|   1 + tan |- - -|   1 + tan |- + -|
        \2   2/           \2   2/           \2   2/
$$\frac{1 - \tan^{2}{\left(\frac{y}{2} + \frac{z}{2} \right)}}{\tan^{2}{\left(\frac{y}{2} + \frac{z}{2} \right)} + 1} - \frac{\tan{\left(\frac{y}{2} + \frac{z}{2} \right)}}{\tan^{2}{\left(\frac{y}{2} + \frac{z}{2} \right)} + 1} + \frac{\tan{\left(\frac{y}{2} - \frac{z}{2} \right)}}{\tan^{2}{\left(\frac{y}{2} - \frac{z}{2} \right)} + 1}$$
       1                  1         
--------------- - ------------------
   /pi        \             /pi    \
csc|-- - y - z|   csc(z)*csc|-- - y|
   \2         /             \2     /
$$\frac{1}{\csc{\left(- y - z + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(z \right)} \csc{\left(- y + \frac{\pi}{2} \right)}}$$
  //     0       for And(im(z) = 0, z mod pi = 0)\                                                                                                                
  ||                                             | //  1     for And(im(y) = 0, y mod 2*pi = 0)\   //    1       for And(im(y) + im(z) = 0, (y + z) mod 2*pi = 0)\
  ||     1                                       | ||                                          |   ||                                                            |
- |<-----------             otherwise            |*|<  1                                       | + |<    1                                                       |
  ||   /    pi\                                  | ||------              otherwise             |   ||----------                     otherwise                    |
  ||sec|z - --|                                  | \\sec(y)                                    /   \\sec(y + z)                                                  /
  \\   \    2 /                                  /                                                                                                                
$$\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(z\right)} = 0 \wedge z \bmod \pi = 0 \\\frac{1}{\sec{\left(z - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(y \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} + \operatorname{im}{\left(z\right)} = 0 \wedge \left(y + z\right) \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(y + z \right)}} & \text{otherwise} \end{cases}\right)$$
    1              1      
---------- - -------------
sec(y + z)   csc(z)*sec(y)
$$\frac{1}{\sec{\left(y + z \right)}} - \frac{1}{\csc{\left(z \right)} \sec{\left(y \right)}}$$
                                     2/pi   y   z\
                                  csc |-- - - - -|
                                      \2    2   2/
                              1 - ----------------
                                       2/y   z\   
                                    csc |- + -|   
     1              1                   \2   2/   
------------ - ------------ + --------------------
2*csc(y - z)   2*csc(y + z)          2/pi   y   z\
                                  csc |-- - - - -|
                                      \2    2   2/
                              1 + ----------------
                                       2/y   z\   
                                    csc |- + -|   
                                        \2   2/   
$$\frac{1 - \frac{\csc^{2}{\left(- \frac{y}{2} - \frac{z}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{y}{2} + \frac{z}{2} \right)}}}{1 + \frac{\csc^{2}{\left(- \frac{y}{2} - \frac{z}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{y}{2} + \frac{z}{2} \right)}}} - \frac{1}{2 \csc{\left(y + z \right)}} + \frac{1}{2 \csc{\left(y - z \right)}}$$
    1                1         
---------- - ------------------
sec(y + z)             /    pi\
             sec(y)*sec|z - --|
                       \    2 /
$$\frac{1}{\sec{\left(y + z \right)}} - \frac{1}{\sec{\left(y \right)} \sec{\left(z - \frac{\pi}{2} \right)}}$$
/       0         for And(-im(z) + im(y) = 0, (y - z) mod pi = 0)   /       0         for And(im(y) + im(z) = 0, (y + z) mod pi = 0)                  
|                                                                   |                                                                                 
|       /y   z\                                                     |       /y   z\                                                                   
|  2*cot|- - -|                                                     |  2*cot|- + -|                                                                   
<       \2   2/                                                     <       \2   2/                                                             1     
|---------------                     otherwise                      |---------------                    otherwise                      1 - -----------
|       2/y   z\                                                    |       2/y   z\                                                          2/y   z\
|1 + cot |- - -|                                                    |1 + cot |- + -|                                                       cot |- + -|
\        \2   2/                                                    \        \2   2/                                                           \2   2/
----------------------------------------------------------------- - ---------------------------------------------------------------- + ---------------
                                2                                                                  2                                            1     
                                                                                                                                       1 + -----------
                                                                                                                                              2/y   z\
                                                                                                                                           cot |- + -|
                                                                                                                                               \2   2/
$$\frac{1 - \frac{1}{\cot^{2}{\left(\frac{y}{2} + \frac{z}{2} \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{y}{2} + \frac{z}{2} \right)}}} + \left(\frac{\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(y\right)} - \operatorname{im}{\left(z\right)} = 0 \wedge \left(y - z\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{y}{2} - \frac{z}{2} \right)}}{\cot^{2}{\left(\frac{y}{2} - \frac{z}{2} \right)} + 1} & \text{otherwise} \end{cases}}{2}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(y\right)} + \operatorname{im}{\left(z\right)} = 0 \wedge \left(y + z\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{y}{2} + \frac{z}{2} \right)}}{\cot^{2}{\left(\frac{y}{2} + \frac{z}{2} \right)} + 1} & \text{otherwise} \end{cases}}{2}\right)$$
                                           2/y   z   pi\
                                        cos |- + - - --|
                                            \2   2   2 /
                                    1 - ----------------
   /        pi\      /        pi\            2/y   z\   
cos|y - z - --|   cos|y + z - --|         cos |- + -|   
   \        2 /      \        2 /             \2   2/   
--------------- - --------------- + --------------------
       2                 2                 2/y   z   pi\
                                        cos |- + - - --|
                                            \2   2   2 /
                                    1 + ----------------
                                             2/y   z\   
                                          cos |- + -|   
                                              \2   2/   
$$\frac{1 - \frac{\cos^{2}{\left(\frac{y}{2} + \frac{z}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{y}{2} + \frac{z}{2} \right)}}}{1 + \frac{\cos^{2}{\left(\frac{y}{2} + \frac{z}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{y}{2} + \frac{z}{2} \right)}}} + \frac{\cos{\left(y - z - \frac{\pi}{2} \right)}}{2} - \frac{\cos{\left(y + z - \frac{\pi}{2} \right)}}{2}$$
                                                                                              //       1         for And(im(y) + im(z) = 0, (y + z) mod 2*pi = 0)\
                                                                                              ||                                                                 |
                                                                                              ||       2/y   z\                                                  |
  //  0     for And(im(z) = 0, z mod pi = 0)\ //  1     for And(im(y) = 0, y mod 2*pi = 0)\   ||1 - tan |- + -|                                                  |
- |<                                        |*|<                                          | + |<        \2   2/                                                  |
  \\sin(z)             otherwise            / \\cos(y)              otherwise             /   ||---------------                     otherwise                    |
                                                                                              ||       2/y   z\                                                  |
                                                                                              ||1 + tan |- + -|                                                  |
                                                                                              \\        \2   2/                                                  /
$$\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(z\right)} = 0 \wedge z \bmod \pi = 0 \\\sin{\left(z \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} = 0 \wedge y \bmod 2 \pi = 0 \\\cos{\left(y \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(y\right)} + \operatorname{im}{\left(z\right)} = 0 \wedge \left(y + z\right) \bmod 2 \pi = 0 \\\frac{1 - \tan^{2}{\left(\frac{y}{2} + \frac{z}{2} \right)}}{\tan^{2}{\left(\frac{y}{2} + \frac{z}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
Раскрыть выражение [src]
cos(y)*cos(z) - cos(y)*sin(z) - sin(y)*sin(z)
$$- \sin{\left(y \right)} \sin{\left(z \right)} - \sin{\left(z \right)} \cos{\left(y \right)} + \cos{\left(y \right)} \cos{\left(z \right)}$$