sin(3*x)*cos(7*x)+cos(3*x)*sin(7*x)еслиx=3 (упростите выражение)

Вы ввели

sin(3*x)*cos(7*x) + cos(3*x)*sin(7*x)

$$\sin{\left(3 x \right)} \cos{\left(7 x \right)} + \sin{\left(7 x \right)} \cos{\left(3 x \right)}$$

Подстановка условия

sin(3*x)*cos(7*x) + cos(3*x)*sin(7*x) при x = 3

подставляем

sin(3*x)*cos(7*x) + cos(3*x)*sin(7*x)

$$\sin{\left(3 x \right)} \cos{\left(7 x \right)} + \sin{\left(7 x \right)} \cos{\left(3 x \right)}$$

sin(10*x)

$$\sin{\left(10 x \right)}$$

переменные

x = 3

$$x = 3$$

sin(10*(3))

$$\sin{\left(10 (3) \right)}$$

sin(10*3)

$$\sin{\left(10 \cdot 3 \right)}$$

sin(30)

$$\sin{\left(30 \right)}$$

Степени

    / -7*I*x    7*I*x\                          / -3*I*x    3*I*x\                     
    |e         e     | /   -3*I*x    3*I*x\     |e         e     | /   -7*I*x    7*I*x\
  I*|------- + ------|*\- e       + e     /   I*|------- + ------|*\- e       + e     /
    \   2        2   /                          \   2        2   /                     
- ----------------------------------------- - -----------------------------------------
                      2                                           2                    

$$- \frac{i \left(\frac{e^{3 i x}}{2} + \frac{e^{- 3 i x}}{2}\right) \left(e^{7 i x} - e^{- 7 i x}\right)}{2} - \frac{i \left(e^{3 i x} - e^{- 3 i x}\right) \left(\frac{e^{7 i x}}{2} + \frac{e^{- 7 i x}}{2}\right)}{2}$$

Численный ответ

cos(3*x)*sin(7*x) + cos(7*x)*sin(3*x)

Общее упрощение

sin(10*x)

$$\sin{\left(10 x \right)}$$

Собрать выражение

sin(10*x)

$$\sin{\left (10 x \right )}$$

Тригонометрическая часть

     /       2/3*x\\    /7*x\          /       2/7*x\\    /3*x\  
   2*|1 - tan |---||*tan|---|        2*|1 - tan |---||*tan|---|  
     \        \ 2 //    \ 2 /          \        \ 2 //    \ 2 /  
------------------------------- + -------------------------------
/       2/3*x\\ /       2/7*x\\   /       2/3*x\\ /       2/7*x\\
|1 + tan |---||*|1 + tan |---||   |1 + tan |---||*|1 + tan |---||
\        \ 2 // \        \ 2 //   \        \ 2 // \        \ 2 //

$$\frac{2 \cdot \left(1 - \tan^{2}{\left(\frac{3 x}{2} \right)}\right) \tan{\left(\frac{7 x}{2} \right)}}{\left(\tan^{2}{\left(\frac{3 x}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{7 x}{2} \right)} + 1\right)} + \frac{2 \cdot \left(1 - \tan^{2}{\left(\frac{7 x}{2} \right)}\right) \tan{\left(\frac{3 x}{2} \right)}}{\left(\tan^{2}{\left(\frac{3 x}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{7 x}{2} \right)} + 1\right)}$$

            /      pi\               /      pi\
cos(3*x)*cos|7*x - --| + cos(7*x)*cos|3*x - --|
            \      2 /               \      2 /

$$\cos{\left(3 x \right)} \cos{\left(7 x - \frac{\pi}{2} \right)} + \cos{\left(7 x \right)} \cos{\left(3 x - \frac{\pi}{2} \right)}$$

          1                        1           
---------------------- + ----------------------
            /      pi\               /      pi\
sec(3*x)*sec|7*x - --|   sec(7*x)*sec|3*x - --|
            \      2 /               \      2 /

$$\frac{1}{\sec{\left(7 x \right)} \sec{\left(3 x - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(3 x \right)} \sec{\left(7 x - \frac{\pi}{2} \right)}}$$

        1                   1        
----------------- + -----------------
csc(3*x)*sec(7*x)   csc(7*x)*sec(3*x)

$$\frac{1}{\csc{\left(7 x \right)} \sec{\left(3 x \right)}} + \frac{1}{\csc{\left(3 x \right)} \sec{\left(7 x \right)}}$$

//      0        for And(im(x) = 0, 3*x mod pi = 0)\                                                     //      0        for And(im(x) = 0, 7*x mod pi = 0)\                                                  
||                                                 | //   1      for And(im(x) = 0, 7*x mod 2*pi = 0)\   ||                                                 | //   1      for And(im(x) = 0, 3*x mod 2*pi = 0)\
||      1                                          | ||                                              |   ||      1                                          | ||                                              |
|<-------------              otherwise             |*|<   1                                          | + |<-------------              otherwise             |*|<   1                                          |
||   /      pi\                                    | ||--------               otherwise              |   ||   /      pi\                                    | ||--------               otherwise              |
||sec|3*x - --|                                    | \\sec(7*x)                                      /   ||sec|7*x - --|                                    | \\sec(3*x)                                      /
\\   \      2 /                                    /                                                     \\   \      2 /                                    /                                                  

$$\left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod \pi = 0 \\\frac{1}{\sec{\left(3 x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 7 x \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(7 x \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 7 x \bmod \pi = 0 \\\frac{1}{\sec{\left(7 x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(3 x \right)}} & \text{otherwise} \end{cases}\right)\right)$$

                                                //      1        for And(im(x) = 0, 7*x mod 2*pi = 0)\                                                   //      1        for And(im(x) = 0, 3*x mod 2*pi = 0)\
//   0      for And(im(x) = 0, 3*x mod pi = 0)\ ||                                                   |   //   0      for And(im(x) = 0, 7*x mod pi = 0)\ ||                                                   |
|<                                            |*|<   /pi      \                                      | + |<                                            |*|<   /pi      \                                      |
\\sin(3*x)              otherwise             / ||sin|-- + 7*x|               otherwise              |   \\sin(7*x)              otherwise             / ||sin|-- + 3*x|               otherwise              |
                                                \\   \2       /                                      /                                                   \\   \2       /                                      /

$$\left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod \pi = 0 \\\sin{\left(3 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 7 x \bmod 2 \pi = 0 \\\sin{\left(7 x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 7 x \bmod \pi = 0 \\\sin{\left(7 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod 2 \pi = 0 \\\sin{\left(3 x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right)$$

          1                        1           
---------------------- + ----------------------
            /pi      \               /pi      \
csc(3*x)*csc|-- - 7*x|   csc(7*x)*csc|-- - 3*x|
            \2       /               \2       /

$$\frac{1}{\csc{\left(7 x \right)} \csc{\left(- 3 x + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(3 x \right)} \csc{\left(- 7 x + \frac{\pi}{2} \right)}}$$

    1    
---------
csc(10*x)

$$\frac{1}{\csc{\left(10 x \right)}}$$

/    0      for And(im(x) = 0, 10*x mod pi = 0)
<                                              
\sin(10*x)               otherwise             

$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 10 x \bmod \pi = 0 \\\sin{\left(10 x \right)} & \text{otherwise} \end{cases}$$

  2*tan(5*x) 
-------------
       2     
1 + tan (5*x)

$$\frac{2 \tan{\left(5 x \right)}}{\tan^{2}{\left(5 x \right)} + 1}$$

//                      0                        for And(im(x) = 0, 3*x mod pi = 0)\ //                       1                         for And(im(x) = 0, 7*x mod 2*pi = 0)\   //                      0                        for And(im(x) = 0, 7*x mod pi = 0)\ //                       1                         for And(im(x) = 0, 3*x mod 2*pi = 0)\
||                                                                                 | ||                                                                                     |   ||                                                                                 | ||                                                                                     |
|

$$\left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod \pi = 0 \\\sin{\left(3 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 7 x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 7 x \bmod 2 \pi = 0 \\\cos{\left(7 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 7 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 7 x \bmod \pi = 0 \\\sin{\left(7 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod 2 \pi = 0 \\\cos{\left(3 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$

//      0        for And(im(x) = 0, 3*x mod pi = 0)\ //      1         for And(im(x) = 0, 7*x mod 2*pi = 0)\   //      0        for And(im(x) = 0, 7*x mod pi = 0)\ //      1         for And(im(x) = 0, 3*x mod 2*pi = 0)\
||                                                 | ||                                                    |   ||                                                 | ||                                                    |
||       /3*x\                                     | ||        2/7*x\                                      |   ||       /7*x\                                     | ||        2/3*x\                                      |
||  2*cot|---|                                     | ||-1 + cot |---|                                      |   ||  2*cot|---|                                     | ||-1 + cot |---|                                      |
|<       \ 2 /                                     |*|<         \ 2 /                                      | + |<       \ 2 /                                     |*|<         \ 2 /                                      |
||-------------              otherwise             | ||--------------               otherwise              |   ||-------------              otherwise             | ||--------------               otherwise              |
||       2/3*x\                                    | ||       2/7*x\                                       |   ||       2/7*x\                                    | ||       2/3*x\                                       |
||1 + cot |---|                                    | ||1 + cot |---|                                       |   ||1 + cot |---|                                    | ||1 + cot |---|                                       |
\\        \ 2 /                                    / \\        \ 2 /                                       /   \\        \ 2 /                                    / \\        \ 2 /                                       /

$$\left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 x}{2} \right)}}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 7 x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{7 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{7 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 7 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{7 x}{2} \right)}}{\cot^{2}{\left(\frac{7 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{3 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$

sin(10*x)

$$\sin{\left(10 x \right)}$$

/      0        for And(im(x) = 0, 10*x mod pi = 0)
|                                                  
|  2*cot(5*x)                                      
<-------------               otherwise             
|       2                                          
|1 + cot (5*x)                                     
\                                                  

$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 10 x \bmod \pi = 0 \\\frac{2 \cot{\left(5 x \right)}}{\cot^{2}{\left(5 x \right)} + 1} & \text{otherwise} \end{cases}$$

//   0      for And(im(x) = 0, 3*x mod pi = 0)\ //   1      for And(im(x) = 0, 7*x mod 2*pi = 0)\   //   0      for And(im(x) = 0, 7*x mod pi = 0)\ //   1      for And(im(x) = 0, 3*x mod 2*pi = 0)\
|<                                            |*|<                                              | + |<                                            |*|<                                              |
\\sin(3*x)              otherwise             / \\cos(7*x)               otherwise              /   \\sin(7*x)              otherwise             / \\cos(3*x)               otherwise              /

$$\left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod \pi = 0 \\\sin{\left(3 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 7 x \bmod 2 \pi = 0 \\\cos{\left(7 x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 7 x \bmod \pi = 0 \\\sin{\left(7 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod 2 \pi = 0 \\\cos{\left(3 x \right)} & \text{otherwise} \end{cases}\right)\right)$$

   /       pi\
cos|10*x - --|
   \       2 /

$$\cos{\left(10 x - \frac{\pi}{2} \right)}$$

            /pi      \               /pi      \
sin(3*x)*sin|-- + 7*x| + sin(7*x)*sin|-- + 3*x|
            \2       /               \2       /

$$\sin{\left(3 x \right)} \sin{\left(7 x + \frac{\pi}{2} \right)} + \sin{\left(7 x \right)} \sin{\left(3 x + \frac{\pi}{2} \right)}$$

      1       
--------------
   /       pi\
sec|10*x - --|
   \       2 /

$$\frac{1}{\sec{\left(10 x - \frac{\pi}{2} \right)}}$$

//                        0                           for And(im(x) = 0, 3*x mod pi = 0)\ //                          1                            for And(im(x) = 0, 7*x mod 2*pi = 0)\   //                        0                           for And(im(x) = 0, 7*x mod pi = 0)\ //                          1                            for And(im(x) = 0, 3*x mod 2*pi = 0)\
||                                                                                      | ||                                                                                           |   ||                                                                                      | ||                                                                                           |
||/      0        for And(im(x) = 0, 3*x mod pi = 0)                                    | ||/      1         for And(im(x) = 0, 7*x mod 2*pi = 0)                                      |   ||/      0        for And(im(x) = 0, 7*x mod pi = 0)                                    | ||/      1         for And(im(x) = 0, 3*x mod 2*pi = 0)                                      |
|||                                                                                     | |||                                                                                          |   |||                                                                                     | |||                                                                                          |
|||       /3*x\                                                                         | |||        2/7*x\                                                                            |   |||       /7*x\                                                                         | |||        2/3*x\                                                                            |
|<|  2*cot|---|                                                                         |*|<|-1 + cot |---|                                                                            | + |<|  2*cot|---|                                                                         |*|<|-1 + cot |---|                                                                            |
||<       \ 2 /                                                   otherwise             | ||<         \ 2 /                                                     otherwise              |   ||<       \ 2 /                                                   otherwise             | ||<         \ 2 /                                                     otherwise              |
|||-------------              otherwise                                                 | |||--------------               otherwise                                                    |   |||-------------              otherwise                                                 | |||--------------               otherwise                                                    |
|||       2/3*x\                                                                        | |||       2/7*x\                                                                             |   |||       2/7*x\                                                                        | |||       2/3*x\                                                                             |
|||1 + cot |---|                                                                        | |||1 + cot |---|                                                                             |   |||1 + cot |---|                                                                        | |||1 + cot |---|                                                                             |
\\\        \ 2 /                                                                        / \\\        \ 2 /                                                                             /   \\\        \ 2 /                                                                        / \\\        \ 2 /                                                                             /

$$\left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 x}{2} \right)}}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 7 x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 7 x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{7 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{7 x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 7 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 7 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{7 x}{2} \right)}}{\cot^{2}{\left(\frac{7 x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{3 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$

//      0        for And(im(x) = 0, 3*x mod pi = 0)\                                                     //      0        for And(im(x) = 0, 7*x mod pi = 0)\                                                  
||                                                 | //   1      for And(im(x) = 0, 7*x mod 2*pi = 0)\   ||                                                 | //   1      for And(im(x) = 0, 3*x mod 2*pi = 0)\
|<   /      pi\                                    |*|<                                              | + |<   /      pi\                                    |*|<                                              |
||cos|3*x - --|              otherwise             | \\cos(7*x)               otherwise              /   ||cos|7*x - --|              otherwise             | \\cos(3*x)               otherwise              /
\\   \      2 /                                    /                                                     \\   \      2 /                                    /                                                  

$$\left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod \pi = 0 \\\cos{\left(3 x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 7 x \bmod 2 \pi = 0 \\\cos{\left(7 x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 7 x \bmod \pi = 0 \\\cos{\left(7 x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod 2 \pi = 0 \\\cos{\left(3 x \right)} & \text{otherwise} \end{cases}\right)\right)$$

//      0        for And(im(x) = 0, 3*x mod pi = 0)\ //      1        for And(im(x) = 0, 7*x mod 2*pi = 0)\   //      0        for And(im(x) = 0, 7*x mod pi = 0)\ //      1        for And(im(x) = 0, 3*x mod 2*pi = 0)\
||                                                 | ||                                                   |   ||                                                 | ||                                                   |
||       /3*x\                                     | ||       2/7*x\                                      |   ||       /7*x\                                     | ||       2/3*x\                                      |
||  2*tan|---|                                     | ||1 - tan |---|                                      |   ||  2*tan|---|                                     | ||1 - tan |---|                                      |
|<       \ 2 /                                     |*|<        \ 2 /                                      | + |<       \ 2 /                                     |*|<        \ 2 /                                      |
||-------------              otherwise             | ||-------------               otherwise              |   ||-------------              otherwise             | ||-------------               otherwise              |
||       2/3*x\                                    | ||       2/7*x\                                      |   ||       2/7*x\                                    | ||       2/3*x\                                      |
||1 + tan |---|                                    | ||1 + tan |---|                                      |   ||1 + tan |---|                                    | ||1 + tan |---|                                      |
\\        \ 2 /                                    / \\        \ 2 /                                      /   \\        \ 2 /                                    / \\        \ 2 /                                      /

$$\left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{3 x}{2} \right)}}{\tan^{2}{\left(\frac{3 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 7 x \bmod 2 \pi = 0 \\\frac{1 - \tan^{2}{\left(\frac{7 x}{2} \right)}}{\tan^{2}{\left(\frac{7 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 7 x \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{7 x}{2} \right)}}{\tan^{2}{\left(\frac{7 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod 2 \pi = 0 \\\frac{1 - \tan^{2}{\left(\frac{3 x}{2} \right)}}{\tan^{2}{\left(\frac{3 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$

                                                //      1        for And(im(x) = 0, 7*x mod 2*pi = 0)\                                                   //      1        for And(im(x) = 0, 3*x mod 2*pi = 0)\
//   0      for And(im(x) = 0, 3*x mod pi = 0)\ ||                                                   |   //   0      for And(im(x) = 0, 7*x mod pi = 0)\ ||                                                   |
||                                            | ||      1                                            |   ||                                            | ||      1                                            |
|<   1                                        |*|<-------------               otherwise              | + |<   1                                        |*|<-------------               otherwise              |
||--------              otherwise             | ||   /pi      \                                      |   ||--------              otherwise             | ||   /pi      \                                      |
\\csc(3*x)                                    / ||csc|-- - 7*x|                                      |   \\csc(7*x)                                    / ||csc|-- - 3*x|                                      |
                                                \\   \2       /                                      /                                                   \\   \2       /                                      /

$$\left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod \pi = 0 \\\frac{1}{\csc{\left(3 x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 7 x \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- 7 x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 7 x \bmod \pi = 0 \\\frac{1}{\csc{\left(7 x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 3 x \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- 3 x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)$$

Раскрыть выражение

         3       3             5                    5                    3       7             7       3                                7                    7                    3                    3                    3       5             5       3   
- 448*cos (x)*sin (x) - 336*cos (x)*sin(x) - 336*sin (x)*cos(x) - 256*cos (x)*sin (x) - 256*cos (x)*sin (x) - 42*cos(x)*sin(x) + 192*cos (x)*sin(x) + 192*sin (x)*cos(x) + 196*cos (x)*sin(x) + 196*sin (x)*cos(x) + 448*cos (x)*sin (x) + 448*cos (x)*sin (x)

$$- 256 \sin^{7}{\left(x \right)} \cos^{3}{\left(x \right)} + 192 \sin^{7}{\left(x \right)} \cos{\left(x \right)} + 448 \sin^{5}{\left(x \right)} \cos^{3}{\left(x \right)} - 336 \sin^{5}{\left(x \right)} \cos{\left(x \right)} - 256 \sin^{3}{\left(x \right)} \cos^{7}{\left(x \right)} + 448 \sin^{3}{\left(x \right)} \cos^{5}{\left(x \right)} - 448 \sin^{3}{\left(x \right)} \cos^{3}{\left(x \right)} + 196 \sin^{3}{\left(x \right)} \cos{\left(x \right)} + 192 \sin{\left(x \right)} \cos^{7}{\left(x \right)} - 336 \sin{\left(x \right)} \cos^{5}{\left(x \right)} + 196 \sin{\left(x \right)} \cos^{3}{\left(x \right)} - 42 \sin{\left(x \right)} \cos{\left(x \right)}$$