sin(a+b)-sin(a)*cos(b)еслиa=-2 (упростите выражение)

Вы ввели

sin(a + b) - sin(a)*cos(b)

$$- \sin{\left(a \right)} \cos{\left(b \right)} + \sin{\left(a + b \right)}$$

Подстановка условия

sin(a + b) - sin(a)*cos(b) при a = -2

подставляем

sin(a + b) - sin(a)*cos(b)

$$- \sin{\left(a \right)} \cos{\left(b \right)} + \sin{\left(a + b \right)}$$

cos(a)*sin(b)

$$\sin{\left(b \right)} \cos{\left(a \right)}$$

переменные

a = -2

$$a = -2$$

cos((-2))*sin(b)

$$\sin{\left(b \right)} \cos{\left((-2) \right)}$$

cos(-2)*sin(b)

$$\sin{\left(b \right)} \cos{\left(-2 \right)}$$

cos(2)*sin(b)

$$\sin{\left(b \right)} \cos{\left(2 \right)}$$

Степени

                                     / I*b    -I*b\                 
                                     |e      e    | /   -I*a    I*a\
    /   I*(-a - b)    I*(a + b)\   I*|---- + -----|*\- e     + e   /
  I*\- e           + e         /     \ 2       2  /                 
- ------------------------------ + ---------------------------------
                2                                  2                

$$\frac{i \left(e^{i a} - e^{- i a}\right) \left(\frac{e^{i b}}{2} + \frac{e^{- i b}}{2}\right)}{2} - \frac{i \left(- e^{i \left(- a - b\right)} + e^{i \left(a + b\right)}\right)}{2}$$

Численный ответ

-cos(b)*sin(a) + sin(a + b)

Общее упрощение

cos(a)*sin(b)

$$\sin{\left(b \right)} \cos{\left(a \right)}$$

Собрать выражение

sin(a + b)   sin(a - b)
---------- - ----------
    2            2     

$$- \frac{1}{2} \sin{\left (a - b \right )} + \frac{1}{2} \sin{\left (a + b \right )}$$

Тригонометрическая часть

cos(a)*sin(b)

$$\sin{\left(b \right)} \cos{\left(a \right)}$$

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

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

              /a   b\                      /a   b\
cos(a + b)*tan|- + -| - cos(b)*sin(a) + tan|- + -|
              \2   2/                      \2   2/

$$- \sin{\left(a \right)} \cos{\left(b \right)} + \cos{\left(a + b \right)} \tan{\left(\frac{a}{2} + \frac{b}{2} \right)} + \tan{\left(\frac{a}{2} + \frac{b}{2} \right)}$$

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

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

                                         2/a + b\         
                                    4*sin |-----|         
  sin(a + b)   sin(a - b)                 \  2  /         
- ---------- - ---------- + ------------------------------
      2            2        /         4/a + b\\           
                            |    4*sin |-----||           
                            |          \  2  /|           
                            |1 + -------------|*sin(a + b)
                            |        2        |           
                            \     sin (a + b) /           

$$- \frac{\sin{\left(a - b \right)}}{2} - \frac{\sin{\left(a + b \right)}}{2} + \frac{4 \sin^{2}{\left(\frac{a + b}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a + b}{2} \right)}}{\sin^{2}{\left(a + b \right)}} + 1\right) \sin{\left(a + b \right)}}$$

sin(a + b)   sin(a - b)
---------- - ----------
    2            2     

$$- \frac{\sin{\left(a - b \right)}}{2} + \frac{\sin{\left(a + b \right)}}{2}$$

            /    pi\      /        pi\
- cos(b)*cos|a - --| + cos|a + b - --|
            \    2 /      \        2 /

$$- \cos{\left(b \right)} \cos{\left(a - \frac{\pi}{2} \right)} + \cos{\left(a + b - \frac{\pi}{2} \right)}$$

    1                1         
---------- - ------------------
csc(a + b)             /pi    \
             csc(a)*csc|-- - b|
                       \2     /

$$\frac{1}{\csc{\left(a + b \right)}} - \frac{1}{\csc{\left(a \right)} \csc{\left(- b + \frac{\pi}{2} \right)}}$$

                                         2/a   b\         
                                    4*sin |- + -|         
  sin(a + b)   sin(a - b)                 \2   2/         
- ---------- - ---------- + ------------------------------
      2            2        /         4/a   b\\           
                            |    4*sin |- + -||           
                            |          \2   2/|           
                            |1 + -------------|*sin(a + b)
                            |        2        |           
                            \     sin (a + b) /           

$$- \frac{\sin{\left(a - b \right)}}{2} - \frac{\sin{\left(a + b \right)}}{2} + \frac{4 \sin^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\sin^{2}{\left(a + b \right)}} + 1\right) \sin{\left(a + b \right)}}$$

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

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

       1                  1         
--------------- - ------------------
   /        pi\             /    pi\
sec|a + b - --|   sec(b)*sec|a - --|
   \        2 /             \    2 /

$$\frac{1}{\sec{\left(a + b - \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(b \right)} \sec{\left(a - \frac{\pi}{2} \right)}}$$

          /    pi\
sin(b)*sin|a + --|
          \    2 /

$$\sin{\left(b \right)} \sin{\left(a + \frac{\pi}{2} \right)}$$

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

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

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

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

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

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

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

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

     /        pi\      /        pi\                /a   b   pi\        
  cos|a + b - --|   cos|a - b - --|           2*cos|- + - - --|        
     \        2 /      \        2 /                \2   2   2 /        
- --------------- - --------------- + ---------------------------------
         2                 2          /       2/a   b   pi\\           
                                      |    cos |- + - - --||           
                                      |        \2   2   2 /|    /a   b\
                                      |1 + ----------------|*cos|- + -|
                                      |         2/a   b\   |    \2   2/
                                      |      cos |- + -|   |           
                                      \          \2   2/   /           

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

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

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

       /a   b\         /       2/b\\    /a\  
  2*tan|- + -|       2*|1 - tan |-||*tan|-|  
       \2   2/         \        \2//    \2/  
--------------- - ---------------------------
       2/a   b\   /       2/a\\ /       2/b\\
1 + tan |- + -|   |1 + tan |-||*|1 + tan |-||
        \2   2/   \        \2// \        \2//

$$- \frac{2 \cdot \left(1 - \tan^{2}{\left(\frac{b}{2} \right)}\right) \tan{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{b}{2} \right)} + 1\right)} + \frac{2 \tan{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1}$$

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

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

  /          2/a\\    /b\    /b\
2*|-1 + 2*cos |-||*cos|-|*sin|-|
  \           \2//    \2/    \2/

$$2 \cdot \left(2 \cos^{2}{\left(\frac{a}{2} \right)} - 1\right) \sin{\left(\frac{b}{2} \right)} \cos{\left(\frac{b}{2} \right)}$$

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

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

            /    pi\             
- sin(a)*sin|b + --| + sin(a + b)
            \    2 /             

$$- \sin{\left(a \right)} \sin{\left(b + \frac{\pi}{2} \right)} + \sin{\left(a + b \right)}$$

                                   /a   b\ 
                              2*tan|- + -| 
  sin(a + b) + sin(a - b)          \2   2/ 
- ----------------------- + ---------------
             2                     2/a   b\
                            1 + tan |- + -|
                                    \2   2/

$$- \frac{\sin{\left(a - b \right)} + \sin{\left(a + b \right)}}{2} + \frac{2 \tan{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1}$$

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

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

                                             /pi   a   b\        
                                        2*csc|-- - - - -|        
       1              1                      \2    2   2/        
- ------------ - ------------ + ---------------------------------
  2*csc(a + b)   2*csc(a - b)   /       2/pi   a   b\\           
                                |    csc |-- - - - -||           
                                |        \2    2   2/|    /a   b\
                                |1 + ----------------|*csc|- + -|
                                |         2/a   b\   |    \2   2/
                                |      csc |- + -|   |           
                                \          \2   2/   /           

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

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

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

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

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

        1         
------------------
          /pi    \
csc(b)*csc|-- - a|
          \2     /

$$\frac{1}{\csc{\left(b \right)} \csc{\left(- a + \frac{\pi}{2} \right)}}$$

  //                    0                      for And(im(a) = 0, a mod pi = 0)\ //                     1                       for And(im(b) = 0, b mod 2*pi = 0)\   //                             0                               for And(im(a) + im(b) = 0, (a + b) mod pi = 0)\
  ||                                                                           | ||                                                                               |   ||                                                                                                           |
- |

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

                                                            /a   b\             
                                                       2*sec|- + -|             
          1                   1                             \2   2/             
- ----------------- - ----------------- + --------------------------------------
       /        pi\        /        pi\   /         2/a   b\   \                
  2*sec|a + b - --|   2*sec|a - b - --|   |      sec |- + -|   |                
       \        2 /        \        2 /   |          \2   2/   |    /a   b   pi\
                                          |1 + ----------------|*sec|- + - - --|
                                          |       2/a   b   pi\|    \2   2   2 /
                                          |    sec |- + - - --||                
                                          \        \2   2   2 //                

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

      /a   b\           /a   b\  
   tan|- + -|        tan|- - -|  
      \2   2/           \2   2/  
--------------- - ---------------
       2/a   b\          2/a   b\
1 + tan |- + -|   1 + tan |- - -|
        \2   2/           \2   2/

$$\frac{\tan{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1} - \frac{\tan{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1}$$

     /       2/a\\    /b\  
   2*|1 - tan |-||*tan|-|  
     \        \2//    \2/  
---------------------------
/       2/a\\ /       2/b\\
|1 + tan |-||*|1 + tan |-||
\        \2// \        \2//

$$\frac{2 \cdot \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right) \tan{\left(\frac{b}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{b}{2} \right)} + 1\right)}$$

    1              1      
---------- - -------------
csc(a + b)   csc(a)*sec(b)

$$\frac{1}{\csc{\left(a + b \right)}} - \frac{1}{\csc{\left(a \right)} \sec{\left(b \right)}}$$

        1         
------------------
          /    pi\
sec(a)*sec|b - --|
          \    2 /

$$\frac{1}{\sec{\left(a \right)} \sec{\left(b - \frac{\pi}{2} \right)}}$$

          /    pi\
cos(a)*cos|b - --|
          \    2 /

$$\cos{\left(a \right)} \cos{\left(b - \frac{\pi}{2} \right)}$$

Раскрыть выражение

cos(a)*sin(b)

$$\sin{\left(b \right)} \cos{\left(a \right)}$$