sin(90-x)еслиx=-1 (упростите выражение)

Вы ввели

sin(90 - x)

$$\sin{\left(90 - x \right)}$$

Подстановка условия

sin(90 - x) при x = -1

подставляем

sin(90 - x)

$$\sin{\left(90 - x \right)}$$

-sin(-90 + x)

$$- \sin{\left(x - 90 \right)}$$

переменные

x = -1

$$x = -1$$

-sin(-90 + (-1))

$$- \sin{\left((-1) - 90 \right)}$$

-sin(-90 - 1)

$$- \sin{\left(-90 - 1 \right)}$$

sin(91)

$$\sin{\left(91 \right)}$$

Степени

-sin(-90 + x)

$$- \sin{\left(x - 90 \right)}$$

   /   I*(-90 + x)    I*(90 - x)\ 
-I*\- e            + e          / 
----------------------------------
                2                 

$$- \frac{i \left(e^{i \left(90 - x\right)} - e^{i \left(x - 90\right)}\right)}{2}$$

Численный ответ

sin(90 - x)

Рациональный знаменатель

-sin(-90 + x)

$$- \sin{\left(x - 90 \right)}$$

Объединение рациональных выражений

-sin(-90 + x)

$$- \sin{\left(x - 90 \right)}$$

Общее упрощение

-sin(-90 + x)

$$- \sin{\left(x - 90 \right)}$$

Собрать выражение

-sin(-90 + x)

$$- \sin{\left (x - 90 \right )}$$

Комбинаторика

-sin(-90 + x)

$$- \sin{\left(x - 90 \right)}$$

Общий знаменатель

-sin(-90 + x)

$$- \sin{\left(x - 90 \right)}$$

Тригонометрическая часть

              /     pi   x\         
        -2*csc|45 + -- - -|         
              \     2    2/         
------------------------------------
/       2/     pi   x\\             
|    csc |45 + -- - -||             
|        \     2    2/|    /      x\
|1 + -----------------|*csc|-45 + -|
|         2/      x\  |    \      2/
|      csc |-45 + -|  |             
\          \      2/  /             

$$- \frac{2 \csc{\left(- \frac{x}{2} + \frac{\pi}{2} + 45 \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} + 45 \right)}}{\csc^{2}{\left(\frac{x}{2} - 45 \right)}}\right) \csc{\left(\frac{x}{2} - 45 \right)}}$$

/        0          for And(im(x) = 0, (90 - x - 28*pi) mod pi = 0)
|                                                                  
|       /      x\                                                  
| -2*tan|-45 + -|                                                  
<       \      2/                                                  
|-----------------                     otherwise                   
|       2/      x\                                                 
|1 + tan |-45 + -|                                                 
\        \      2/                                                 

$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \left(- x - 28 \pi + 90\right) \bmod \pi = 0 \\- \frac{2 \tan{\left(\frac{x}{2} - 45 \right)}}{\tan^{2}{\left(\frac{x}{2} - 45 \right)} + 1} & \text{otherwise} \end{cases}$$

               2/      x\         
         -4*sin |-45 + -|         
                \      2/         
----------------------------------
/         4/      x\\             
|    4*sin |-45 + -||             
|          \      2/|             
|1 + ---------------|*sin(-90 + x)
|        2          |             
\     sin (-90 + x) /             

$$- \frac{4 \sin^{2}{\left(\frac{x}{2} - 45 \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} - 45 \right)}}{\sin^{2}{\left(x - 90 \right)}} + 1\right) \sin{\left(x - 90 \right)}}$$

/                                   0                                     for And(im(x) = 0, (90 - x - 28*pi) mod pi = 0)
|                                                                                                                        
| //        0          for And(im(x) = 0, (-90 + x + 29*pi) mod pi = 0)\                                                 
| ||                                                                   |                                                 
| ||       /      x\                                                   |                                                 
< ||  2*cot|-45 + -|                                                   |                                                 
|-|<       \      2/                                                   |                     otherwise                   
| ||-----------------                     otherwise                    |                                                 
| ||       2/      x\                                                  |                                                 
| ||1 + cot |-45 + -|                                                  |                                                 
\ \\        \      2/                                                  /                                                 

$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \left(- x - 28 \pi + 90\right) \bmod \pi = 0 \\- \begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \left(x - 90 + 29 \pi\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} - 45 \right)}}{\cot^{2}{\left(\frac{x}{2} - 45 \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}$$

       -1        
-----------------
   /          pi\
sec|-90 + x - --|
   \          2 /

$$- \frac{1}{\sec{\left(x - 90 - \frac{\pi}{2} \right)}}$$

      /      x\    /      x\
-2*sec|-45 + -|*sin|-45 + -|
      \      2/    \      2/
----------------------------
            2/      x\      
     1 + tan |-45 + -|      
             \      2/      

$$- \frac{2 \sin{\left(\frac{x}{2} - 45 \right)} \sec{\left(\frac{x}{2} - 45 \right)}}{\tan^{2}{\left(\frac{x}{2} - 45 \right)} + 1}$$

 //        0          for And(im(x) = 0, (-90 + x + 29*pi) mod pi = 0)\
 ||                                                                   |
 ||       /      x\                                                   |
 ||  2*cot|-45 + -|                                                   |
-|<       \      2/                                                   |
 ||-----------------                     otherwise                    |
 ||       2/      x\                                                  |
 ||1 + cot |-45 + -|                                                  |
 \\        \      2/                                                  /

$$- \begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \left(x - 90 + 29 \pi\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} - 45 \right)}}{\cot^{2}{\left(\frac{x}{2} - 45 \right)} + 1} & \text{otherwise} \end{cases}$$

    /          pi\
-cos|-90 + x - --|
    \          2 /

$$- \cos{\left(x - 90 - \frac{\pi}{2} \right)}$$

-sin(-90 + x)

$$- \sin{\left(x - 90 \right)}$$

/        0          for And(im(x) = 0, (90 - x - 28*pi) mod pi = 0)
|                                                                  
|       /      x\                                                  
| -2*cot|-45 + -|                                                  
<       \      2/                                                  
|-----------------                     otherwise                   
|       2/      x\                                                 
|1 + cot |-45 + -|                                                 
\        \      2/                                                 

$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \left(- x - 28 \pi + 90\right) \bmod \pi = 0 \\- \frac{2 \cot{\left(\frac{x}{2} - 45 \right)}}{\cot^{2}{\left(\frac{x}{2} - 45 \right)} + 1} & \text{otherwise} \end{cases}$$

/                                0                                   for And(im(x) = 0, (90 - x - 28*pi) mod pi = 0)
|                                                                                                                   
< //     0        for And(im(x) = 0, (-90 + x + 29*pi) mod pi = 0)\                                                 
|-|<                                                              |                     otherwise                   
\ \\sin(-90 + x)                     otherwise                    /                                                 

$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \left(- x - 28 \pi + 90\right) \bmod \pi = 0 \\- \begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \left(x - 90 + 29 \pi\right) \bmod \pi = 0 \\\sin{\left(x - 90 \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}$$

   /         pi\
cos|90 - x - --|
   \         2 /

$$\cos{\left(- x - \frac{\pi}{2} + 90 \right)}$$

 //     0        for And(im(x) = 0, (-90 + x + 29*pi) mod pi = 0)\
-|<                                                              |
 \\sin(-90 + x)                     otherwise                    /

$$- \begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \left(x - 90 + 29 \pi\right) \bmod \pi = 0 \\\sin{\left(x - 90 \right)} & \text{otherwise} \end{cases}$$

/        0          for And(im(x) = 0, (90 - x - 28*pi) mod pi = 0)
|                                                                  
|       -1                                                         
<-----------------                     otherwise                   
|   /          pi\                                                 
|sec|-90 + x - --|                                                 
\   \          2 /                                                 

$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \left(- x - 28 \pi + 90\right) \bmod \pi = 0 \\- \frac{1}{\sec{\left(x - 90 - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}$$

/     0        for And(im(x) = 0, (90 - x - 28*pi) mod pi = 0)
|                                                             
<    -1                                                       
|------------                     otherwise                   
\csc(-90 + x)                                                 

$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \left(- x - 28 \pi + 90\right) \bmod \pi = 0 \\- \frac{1}{\csc{\left(x - 90 \right)}} & \text{otherwise} \end{cases}$$

/        0           for And(im(x) = 0, (90 - x - 28*pi) mod pi = 0)
|                                                                   
<    /          pi\                                                 
|-cos|-90 + x - --|                     otherwise                   
\    \          2 /                                                 

$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \left(- x - 28 \pi + 90\right) \bmod \pi = 0 \\- \cos{\left(x - 90 - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}$$

       1        
----------------
   /         pi\
sec|90 - x - --|
   \         2 /

$$\frac{1}{\sec{\left(- x - \frac{\pi}{2} + 90 \right)}}$$

/      0        for And(im(x) = 0, (90 - x - 28*pi) mod pi = 0)
<                                                              
\-sin(-90 + x)                     otherwise                   

$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \left(- x - 28 \pi + 90\right) \bmod \pi = 0 \\- \sin{\left(x - 90 \right)} & \text{otherwise} \end{cases}$$

              -2                
--------------------------------
/          1      \    /      x\
|1 + -------------|*cot|-45 + -|
|       2/      x\|    \      2/
|    cot |-45 + -||             
\        \      2//             

$$- \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} - 45 \right)}}\right) \cot{\left(\frac{x}{2} - 45 \right)}}$$

               /      x   pi\        
         -2*cos|-45 + - - --|        
               \      2   2 /        
-------------------------------------
/       2/      x   pi\\             
|    cos |-45 + - - --||             
|        \      2   2 /|    /      x\
|1 + ------------------|*cos|-45 + -|
|         2/      x\   |    \      2/
|      cos |-45 + -|   |             
\          \      2/   /             

$$- \frac{2 \cos{\left(\frac{x}{2} - 45 - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - 45 - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - 45 \right)}}\right) \cos{\left(\frac{x}{2} - 45 \right)}}$$

     /      x\                   /      x\
- tan|-45 + -| - cos(-90 + x)*tan|-45 + -|
     \      2/                   \      2/

$$- \cos{\left(x - 90 \right)} \tan{\left(\frac{x}{2} - 45 \right)} - \tan{\left(\frac{x}{2} - 45 \right)}$$

                   /      x\              
             -2*sec|-45 + -|              
                   \      2/              
------------------------------------------
/         2/      x\   \                  
|      sec |-45 + -|   |                  
|          \      2/   |    /      x   pi\
|1 + ------------------|*sec|-45 + - - --|
|       2/      x   pi\|    \      2   2 /
|    sec |-45 + - - --||                  
\        \      2   2 //                  

$$- \frac{2 \sec{\left(\frac{x}{2} - 45 \right)}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} - 45 \right)}}{\sec^{2}{\left(\frac{x}{2} - 45 - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{x}{2} - 45 - \frac{\pi}{2} \right)}}$$

    -1      
------------
csc(-90 + x)

$$- \frac{1}{\csc{\left(x - 90 \right)}}$$

       /      x\ 
 -2*tan|-45 + -| 
       \      2/ 
-----------------
       2/      x\
1 + tan |-45 + -|
        \      2/

$$- \frac{2 \tan{\left(\frac{x}{2} - 45 \right)}}{\tan^{2}{\left(\frac{x}{2} - 45 \right)} + 1}$$

Раскрыть выражение

cos(x)*sin(90) - cos(90)*sin(x)

$$- \sin{\left(x \right)} \cos{\left(90 \right)} + \sin{\left(90 \right)} \cos{\left(x \right)}$$