Найти значение выражения (2*sin(a)+3*cos(a))/(3*sin(a)-5*cos(a))еслиa=2 ((2 умножить на синус от (a) плюс 3 умножить на косинус от (a)) делить на (3 умножить на синус от (a) минус 5 умножить на косинус от (a))еслиa равно 2) [Есть ответ!]

(2*sin(a)+3*cos(a))/(3*sin(a)-5*cos(a))еслиa=2 (упростите выражение)

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Решение

Вы ввели [src]
2*sin(a) + 3*cos(a)
-------------------
3*sin(a) - 5*cos(a)
$$\frac{2 \sin{\left(a \right)} + 3 \cos{\left(a \right)}}{3 \sin{\left(a \right)} - 5 \cos{\left(a \right)}}$$
Подстановка условия [src]
(2*sin(a) + 3*cos(a))/(3*sin(a) - 5*cos(a)) при a = 2
подставляем
2*sin(a) + 3*cos(a)
-------------------
3*sin(a) - 5*cos(a)
$$\frac{2 \sin{\left(a \right)} + 3 \cos{\left(a \right)}}{3 \sin{\left(a \right)} - 5 \cos{\left(a \right)}}$$
2*sin(a) + 3*cos(a) 
--------------------
-5*cos(a) + 3*sin(a)
$$\frac{2 \sin{\left(a \right)} + 3 \cos{\left(a \right)}}{3 \sin{\left(a \right)} - 5 \cos{\left(a \right)}}$$
переменные
a = 2
$$a = 2$$
2*sin((2)) + 3*cos((2)) 
------------------------
-5*cos((2)) + 3*sin((2))
$$\frac{2 \sin{\left((2) \right)} + 3 \cos{\left((2) \right)}}{3 \sin{\left((2) \right)} - 5 \cos{\left((2) \right)}}$$
2*sin(2) + 3*cos(2) 
--------------------
-5*cos(2) + 3*sin(2)
$$\frac{3 \cos{\left(2 \right)} + 2 \sin{\left(2 \right)}}{- 5 \cos{\left(2 \right)} + 3 \sin{\left(2 \right)}}$$
Степени [src]
     I*a      -I*a                       
  3*e      3*e         /   -I*a    I*a\  
  ------ + ------- - I*\- e     + e   /  
    2         2                          
-----------------------------------------
     I*a      -I*a       /   -I*a    I*a\
  5*e      5*e       3*I*\- e     + e   /
- ------ - ------- - --------------------
    2         2               2          
$$\frac{- i \left(e^{i a} - e^{- i a}\right) + \frac{3 e^{i a}}{2} + \frac{3 e^{- i a}}{2}}{- \frac{3 i \left(e^{i a} - e^{- i a}\right)}{2} - \frac{5 e^{i a}}{2} - \frac{5 e^{- i a}}{2}}$$
Численный ответ [src]
(2.0*sin(a) + 3.0*cos(a))/(3.0*sin(a) - 5.0*cos(a))
Рациональный знаменатель [src]
      2*sin(a)               3*cos(a)      
-------------------- + --------------------
-5*cos(a) + 3*sin(a)   -5*cos(a) + 3*sin(a)
$$\frac{2 \sin{\left(a \right)}}{3 \sin{\left(a \right)} - 5 \cos{\left(a \right)}} + \frac{3 \cos{\left(a \right)}}{3 \sin{\left(a \right)} - 5 \cos{\left(a \right)}}$$
Собрать выражение [src]
      2*sin(a)               3*cos(a)      
-------------------- + --------------------
-5*cos(a) + 3*sin(a)   -5*cos(a) + 3*sin(a)
$$\frac{2 \sin{\left (a \right )}}{3 \sin{\left (a \right )} - 5 \cos{\left (a \right )}} + \frac{3 \cos{\left (a \right )}}{3 \sin{\left (a \right )} - 5 \cos{\left (a \right )}}$$
Общий знаменатель [src]
  3         19*sin(a)       
- - - ----------------------
  5   -15*sin(a) + 25*cos(a)
$$- \frac{3}{5} - \frac{19 \sin{\left(a \right)}}{- 15 \sin{\left(a \right)} + 25 \cos{\left(a \right)}}$$
Тригонометрическая часть [src]
      2          3    
 ----------- + ------ 
    /    pi\   sec(a) 
 sec|a - --|          
    \    2 /          
----------------------
    5           3     
- ------ + -----------
  sec(a)      /    pi\
           sec|a - --|
              \    2 /
$$\frac{\frac{2}{\sec{\left(a - \frac{\pi}{2} \right)}} + \frac{3}{\sec{\left(a \right)}}}{\frac{3}{\sec{\left(a - \frac{\pi}{2} \right)}} - \frac{5}{\sec{\left(a \right)}}}$$
      /    pi\           
 2*cos|a - --| + 3*cos(a)
      \    2 /           
-------------------------
                 /    pi\
-5*cos(a) + 3*cos|a - --|
                 \    2 /
$$\frac{3 \cos{\left(a \right)} + 2 \cos{\left(a - \frac{\pi}{2} \right)}}{- 5 \cos{\left(a \right)} + 3 \cos{\left(a - \frac{\pi}{2} \right)}}$$
   //     0       for And(im(a) = 0, a mod pi = 0)\     //     1        for And(im(a) = 0, a mod 2*pi = 0)\ 
   ||                                             |     ||                                                | 
   ||       /a\                                   |     ||        2/a\                                    | 
   ||  2*cot|-|                                   |     ||-1 + cot |-|                                    | 
 2*|<       \2/                                   | + 3*|<         \2/                                    | 
   ||-----------             otherwise            |     ||------------              otherwise             | 
   ||       2/a\                                  |     ||       2/a\                                     | 
   ||1 + cot |-|                                  |     ||1 + cot |-|                                     | 
   \\        \2/                                  /     \\        \2/                                     / 
------------------------------------------------------------------------------------------------------------
    //     1        for And(im(a) = 0, a mod 2*pi = 0)\     //     0       for And(im(a) = 0, a mod pi = 0)\
    ||                                                |     ||                                             |
    ||        2/a\                                    |     ||       /a\                                   |
    ||-1 + cot |-|                                    |     ||  2*cot|-|                                   |
- 5*|<         \2/                                    | + 3*|<       \2/                                   |
    ||------------              otherwise             |     ||-----------             otherwise            |
    ||       2/a\                                     |     ||       2/a\                                  |
    ||1 + cot |-|                                     |     ||1 + cot |-|                                  |
    \\        \2/                                     /     \\        \2/                                  /
$$\frac{\left(2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)}{\left(3 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)}$$
   //     0       for And(im(a) = 0, a mod pi = 0)\     //     1       for And(im(a) = 0, a mod 2*pi = 0)\ 
   ||                                             |     ||                                               | 
   ||       /a\                                   |     ||       2/a\                                    | 
   ||  2*tan|-|                                   |     ||1 - tan |-|                                    | 
 2*|<       \2/                                   | + 3*|<        \2/                                    | 
   ||-----------             otherwise            |     ||-----------              otherwise             | 
   ||       2/a\                                  |     ||       2/a\                                    | 
   ||1 + tan |-|                                  |     ||1 + tan |-|                                    | 
   \\        \2/                                  /     \\        \2/                                    / 
-----------------------------------------------------------------------------------------------------------
    //     1       for And(im(a) = 0, a mod 2*pi = 0)\     //     0       for And(im(a) = 0, a mod pi = 0)\
    ||                                               |     ||                                             |
    ||       2/a\                                    |     ||       /a\                                   |
    ||1 - tan |-|                                    |     ||  2*tan|-|                                   |
- 5*|<        \2/                                    | + 3*|<       \2/                                   |
    ||-----------              otherwise             |     ||-----------             otherwise            |
    ||       2/a\                                    |     ||       2/a\                                  |
    ||1 + tan |-|                                    |     ||1 + tan |-|                                  |
    \\        \2/                                    /     \\        \2/                                  /
$$\frac{\left(2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{1 - \tan^{2}{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)}{\left(3 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{1 - \tan^{2}{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)}$$
   2           3      
 ------ + ----------- 
 csc(a)      /pi    \ 
          csc|-- - a| 
             \2     / 
----------------------
       5          3   
- ----------- + ------
     /pi    \   csc(a)
  csc|-- - a|         
     \2     /         
$$\frac{\frac{3}{\csc{\left(- a + \frac{\pi}{2} \right)}} + \frac{2}{\csc{\left(a \right)}}}{- \frac{5}{\csc{\left(- a + \frac{\pi}{2} \right)}} + \frac{3}{\csc{\left(a \right)}}}$$
                 /    pi\ 
 2*sin(a) + 3*sin|a + --| 
                 \    2 / 
--------------------------
       /    pi\           
- 5*sin|a + --| + 3*sin(a)
       \    2 /           
$$\frac{2 \sin{\left(a \right)} + 3 \sin{\left(a + \frac{\pi}{2} \right)}}{3 \sin{\left(a \right)} - 5 \sin{\left(a + \frac{\pi}{2} \right)}}$$
                                                   //     1       for And(im(a) = 0, a mod 2*pi = 0)\ 
   //  0     for And(im(a) = 0, a mod pi = 0)\     ||                                               | 
   ||                                        |     ||     1                                         | 
 2*|<  1                                     | + 3*|<-----------              otherwise             | 
   ||------             otherwise            |     ||   /pi    \                                    | 
   \\csc(a)                                  /     ||csc|-- - a|                                    | 
                                                   \\   \2     /                                    / 
------------------------------------------------------------------------------------------------------
    //     1       for And(im(a) = 0, a mod 2*pi = 0)\                                                
    ||                                               |     //  0     for And(im(a) = 0, a mod pi = 0)\
    ||     1                                         |     ||                                        |
- 5*|<-----------              otherwise             | + 3*|<  1                                     |
    ||   /pi    \                                    |     ||------             otherwise            |
    ||csc|-- - a|                                    |     \\csc(a)                                  /
    \\   \2     /                                    /                                                
$$\frac{\left(2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1}{\csc{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)}{\left(3 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1}{\csc{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)}$$
   //     0       for And(im(a) = 0, a mod pi = 0)\                                                   
   ||                                             |     //  1     for And(im(a) = 0, a mod 2*pi = 0)\ 
   ||     1                                       |     ||                                          | 
 2*|<-----------             otherwise            | + 3*|<  1                                       | 
   ||   /    pi\                                  |     ||------              otherwise             | 
   ||sec|a - --|                                  |     \\sec(a)                                    / 
   \\   \    2 /                                  /                                                   
------------------------------------------------------------------------------------------------------
                                                      //     0       for And(im(a) = 0, a mod pi = 0)\
    //  1     for And(im(a) = 0, a mod 2*pi = 0)\     ||                                             |
    ||                                          |     ||     1                                       |
- 5*|<  1                                       | + 3*|<-----------             otherwise            |
    ||------              otherwise             |     ||   /    pi\                                  |
    \\sec(a)                                    /     ||sec|a - --|                                  |
                                                      \\   \    2 /                                  /
$$\frac{\left(2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right)}{\left(3 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right)}$$
   //                    0                      for And(im(a) = 0, a mod pi = 0)\     //                     1                       for And(im(a) = 0, a mod 2*pi = 0)\ 
   ||                                                                           |     ||                                                                               | 
 2*|
$$\frac{\left(2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)}{\left(3 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)}$$
   //     0       for And(im(a) = 0, a mod pi = 0)\                                                   
   ||                                             |     //  1     for And(im(a) = 0, a mod 2*pi = 0)\ 
 2*|<   /    pi\                                  | + 3*|<                                          | 
   ||cos|a - --|             otherwise            |     \\cos(a)              otherwise             / 
   \\   \    2 /                                  /                                                   
------------------------------------------------------------------------------------------------------
                                                      //     0       for And(im(a) = 0, a mod pi = 0)\
    //  1     for And(im(a) = 0, a mod 2*pi = 0)\     ||                                             |
- 5*|<                                          | + 3*|<   /    pi\                                  |
    \\cos(a)              otherwise             /     ||cos|a - --|             otherwise            |
                                                      \\   \    2 /                                  /
$$\frac{\left(2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\cos{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right)}{\left(3 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\cos{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right)}$$
   /       2/a\\          /a\  
 3*|1 - tan |-||     4*tan|-|  
   \        \2//          \2/  
 --------------- + ----------- 
          2/a\            2/a\ 
   1 + tan |-|     1 + tan |-| 
           \2/             \2/ 
-------------------------------
    /       2/a\\          /a\ 
  5*|1 - tan |-||     6*tan|-| 
    \        \2//          \2/ 
- --------------- + -----------
           2/a\            2/a\
    1 + tan |-|     1 + tan |-|
            \2/             \2/
$$\frac{\frac{3 \cdot \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{4 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}}{- \frac{5 \cdot \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{6 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}}$$
                                                   //     1       for And(im(a) = 0, a mod 2*pi = 0)\ 
   //  0     for And(im(a) = 0, a mod pi = 0)\     ||                                               | 
 2*|<                                        | + 3*|<   /    pi\                                    | 
   \\sin(a)             otherwise            /     ||sin|a + --|              otherwise             | 
                                                   \\   \    2 /                                    / 
------------------------------------------------------------------------------------------------------
    //     1       for And(im(a) = 0, a mod 2*pi = 0)\                                                
    ||                                               |     //  0     for And(im(a) = 0, a mod pi = 0)\
- 5*|<   /    pi\                                    | + 3*|<                                        |
    ||sin|a + --|              otherwise             |     \\sin(a)             otherwise            /
    \\   \    2 /                                    /                                                
$$\frac{\left(2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\sin{\left(a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right)}{\left(3 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\sin{\left(a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right)}$$
   2        3    
 ------ + ------ 
 csc(a)   sec(a) 
-----------------
    5        3   
- ------ + ------
  sec(a)   csc(a)
$$\frac{\frac{3}{\sec{\left(a \right)}} + \frac{2}{\csc{\left(a \right)}}}{- \frac{5}{\sec{\left(a \right)}} + \frac{3}{\csc{\left(a \right)}}}$$
   //                      0                         for And(im(a) = 0, a mod pi = 0)\     //                        1                          for And(im(a) = 0, a mod 2*pi = 0)\ 
   ||                                                                                |     ||                                                                                     | 
   ||/     0       for And(im(a) = 0, a mod pi = 0)                                  |     ||/     1        for And(im(a) = 0, a mod 2*pi = 0)                                    | 
   |||                                                                               |     |||                                                                                    | 
   |||       /a\                                                                     |     |||        2/a\                                                                        | 
 2*|<|  2*cot|-|                                                                     | + 3*|<|-1 + cot |-|                                                                        | 
   ||<       \2/                                                otherwise            |     ||<         \2/                                                  otherwise             | 
   |||-----------             otherwise                                              |     |||------------              otherwise                                                 | 
   |||       2/a\                                                                    |     |||       2/a\                                                                         | 
   |||1 + cot |-|                                                                    |     |||1 + cot |-|                                                                         | 
   \\\        \2/                                                                    /     \\\        \2/                                                                         / 
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
    //                        1                          for And(im(a) = 0, a mod 2*pi = 0)\     //                      0                         for And(im(a) = 0, a mod pi = 0)\
    ||                                                                                     |     ||                                                                                |
    ||/     1        for And(im(a) = 0, a mod 2*pi = 0)                                    |     ||/     0       for And(im(a) = 0, a mod pi = 0)                                  |
    |||                                                                                    |     |||                                                                               |
    |||        2/a\                                                                        |     |||       /a\                                                                     |
- 5*|<|-1 + cot |-|                                                                        | + 3*|<|  2*cot|-|                                                                     |
    ||<         \2/                                                  otherwise             |     ||<       \2/                                                otherwise            |
    |||------------              otherwise                                                 |     |||-----------             otherwise                                              |
    |||       2/a\                                                                         |     |||       2/a\                                                                    |
    |||1 + cot |-|                                                                         |     |||1 + cot |-|                                                                    |
    \\\        \2/                                                                         /     \\\        \2/                                                                    /
$$\frac{\left(2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)}{\left(3 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)}$$
   //  0     for And(im(a) = 0, a mod pi = 0)\     //  1     for And(im(a) = 0, a mod 2*pi = 0)\ 
 2*|<                                        | + 3*|<                                          | 
   \\sin(a)             otherwise            /     \\cos(a)              otherwise             / 
-------------------------------------------------------------------------------------------------
    //  1     for And(im(a) = 0, a mod 2*pi = 0)\     //  0     for And(im(a) = 0, a mod pi = 0)\
- 5*|<                                          | + 3*|<                                        |
    \\cos(a)              otherwise             /     \\sin(a)             otherwise            /
$$\frac{\left(2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right)}{\left(3 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right)}$$
Комбинаторика [src]
-(2*sin(a) + 3*cos(a)) 
-----------------------
  -3*sin(a) + 5*cos(a) 
$$- \frac{2 \sin{\left(a \right)} + 3 \cos{\left(a \right)}}{- 3 \sin{\left(a \right)} + 5 \cos{\left(a \right)}}$$
Раскрыть выражение [src]
      2*sin(a)               3*cos(a)      
-------------------- + --------------------
-5*cos(a) + 3*sin(a)   -5*cos(a) + 3*sin(a)
$$\frac{2 \sin{\left(a \right)}}{3 \sin{\left(a \right)} - 5 \cos{\left(a \right)}} + \frac{3 \cos{\left(a \right)}}{3 \sin{\left(a \right)} - 5 \cos{\left(a \right)}}$$