sin(a)*cos(a)*cot(a)-1еслиa=-2 (упростите выражение)

Вы ввели

sin(a)*cos(a)*cot(a) - 1

$$\sin{\left(a \right)} \cos{\left(a \right)} \cot{\left(a \right)} - 1$$

Подстановка условия

sin(a)*cos(a)*cot(a) - 1*1 при a = -2

подставляем

sin(a)*cos(a)*cot(a) - 1

$$\sin{\left(a \right)} \cos{\left(a \right)} \cot{\left(a \right)} - 1$$

    2   
-sin (a)

$$- \sin^{2}{\left(a \right)}$$

переменные

a = -2

$$a = -2$$

    2      
-sin ((-2))

$$- \sin^{2}{\left((-2) \right)}$$

    2    
-sin (-2)

$$- \sin^{2}{\left(-2 \right)}$$

    2   
-sin (2)

$$- \sin^{2}{\left(2 \right)}$$

Степени

       / I*a    -I*a\                        
       |e      e    | /   -I*a    I*a\       
     I*|---- + -----|*\- e     + e   /*cot(a)
       \ 2       2  /                        
-1 - ----------------------------------------
                        2                    

$$- \frac{i \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right) \left(e^{i a} - e^{- i a}\right) \cot{\left(a \right)}}{2} - 1$$

Численный ответ

-1.0 + cos(a)*cot(a)*sin(a)

Общее упрощение

    2   
-sin (a)

$$- \sin^{2}{\left(a \right)}$$

Собрать выражение

  1   cos(2*a)
- - + --------
  2      2    

$$\frac{1}{2} \cos{\left (2 a \right )} - \frac{1}{2}$$

Тригонометрическая часть

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

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

    2/    pi\
-cos |a - --|
     \    2 /

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

    -1      
------------
   2/    pi\
sec |a - --|
    \    2 /

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

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

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

     //                      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\                                                                        |       
-1 + |<|  2*cot|-|                                                                     |*|<|-1 + cot |-|                                                                        |*cot(a)
     ||<       \2/                                                otherwise            | ||<         \2/                                                  otherwise             |       
     |||-----------             otherwise                                              | |||------------              otherwise                                                 |       
     |||       2/a\                                                                    | |||       2/a\                                                                         |       
     |||1 + cot |-|                                                                    | |||1 + cot |-|                                                                         |       
     \\\        \2/                                                                    / \\\        \2/                                                                         /       

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

     cos(a)*sin(2*a)
-1 + ---------------
         2*sin(a)   

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

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

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

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

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

     sin(2*a)
-1 + --------
     2*tan(a)

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

 //      0         for And(im(a) = 0, a mod pi = 0)\
 ||                                                |
 ||       2/a\                                     |
 ||  4*cot |-|                                     |
 ||        \2/                                     |
-|<--------------             otherwise            |
 ||             2                                  |
 ||/       2/a\\                                   |
 |||1 + cot |-||                                   |
 ||\        \2//                                   |
 \\                                                /

$$- \begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}$$

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

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

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

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

    2   
-sin (a)

$$- \sin^{2}{\left(a \right)}$$

     //     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/                                    |
     ||-----------             otherwise            | ||-----------              otherwise             |
     ||       2/a\                                  | ||       2/a\                                    |
     ||1 + tan |-|                                  | ||1 + tan |-|                                    |
     \\        \2/                                  / \\        \2/                                    /
-1 + ---------------------------------------------------------------------------------------------------
                                                    tan(a)                                              

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

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

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

     //     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 |-|                                    |       
-1 + |<       \2/                                   |*|<         \2/                                    |*cot(a)
     ||-----------             otherwise            | ||------------              otherwise             |       
     ||       2/a\                                  | ||       2/a\                                     |       
     ||1 + cot |-|                                  | ||1 + cot |-|                                     |       
     \\        \2/                                  / \\        \2/                                     /       

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

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

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

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

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

        2   
-1 + cos (a)

$$\cos^{2}{\left(a \right)} - 1$$

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

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

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

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

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

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

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

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

          1      
-1 + ------------
        2/pi    \
     csc |-- - a|
         \2     /

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

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

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

          1     
-1 + -----------
            2   
     1 + tan (a)

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

  1   cos(2*a)
- - + --------
  2      2    

$$\frac{\cos{\left(2 a \right)}}{2} - \frac{1}{2}$$

  -1   
-------
   2   
csc (a)

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

        1   
-1 + -------
        2   
     sec (a)

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

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

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

        2     
     sin (2*a)
-1 + ---------
          2   
     4*sin (a)

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

        2/a\  
  -4*tan |-|  
         \2/  
--------------
             2
/       2/a\\ 
|1 + tan |-|| 
\        \2// 

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