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Simplificación de expresiones/ Descomposición en fracciones simples/ sinx/(n*sin(2x-2arcsin(sinx/n)))

¿Cómo vas a descomponer esta sinx/(n*sin(2x-2arcsin(sinx/n))) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
           sin(x)          
---------------------------
     /            /sin(x)\\
n*sin|2*x - 2*asin|------||
     \            \  n   //
sin⁡(x)nsin⁡(2x−2asin⁡(sin⁡(x)n))\frac{\sin{\left(x \right)}}{n \sin{\left(2 x - 2 \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)} \right)}} 
sin(x)/((n*sin(2*x - 2*asin(sin(x)/n))))
Respuesta numérica [src] 
sin(x)/(n*sin(2*x - 2*asin(sin(x)/n)))
sin(x)/(n*sin(2*x - 2*asin(sin(x)/n)))
Unión de expresiones racionales [src] 
           sin(x)          
---------------------------
     /  /        /sin(x)\\\
n*sin|2*|x - asin|------|||
     \  \        \  n   ///
sin⁡(x)nsin⁡(2(x−asin⁡(sin⁡(x)n)))\frac{\sin{\left(x \right)}}{n \sin{\left(2 \left(x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)}\right) \right)}} 
sin(x)/(n*sin(2*(x - asin(sin(x)/n))))
Potencias [src] 
                                      -I*x    I*x                                   
                                   - e     + e                                      
------------------------------------------------------------------------------------
  /     /             /  /   -I*x    I*x\\\      /            /  /   -I*x    I*x\\\\
  |     |             |I*\- e     + e   /||      |            |I*\- e     + e   /|||
  |   I*|-2*x - 2*asin|------------------||    I*|2*x + 2*asin|------------------|||
  |     \             \       2*n        //      \            \       2*n        //|
n*\- e                                      + e                                    /
eix−e−ixn(−ei(−2x−2asin⁡(i(eix−e−ix)2n))+ei(2x+2asin⁡(i(eix−e−ix)2n)))\frac{e^{i x} - e^{- i x}}{n \left(- e^{i \left(- 2 x - 2 \operatorname{asin}{\left(\frac{i \left(e^{i x} - e^{- i x}\right)}{2 n} \right)}\right)} + e^{i \left(2 x + 2 \operatorname{asin}{\left(\frac{i \left(e^{i x} - e^{- i x}\right)}{2 n} \right)}\right)}\right)} 
(-exp(-i*x) + exp(i*x))/(n*(-exp(i*(-2*x - 2*asin(i*(-exp(-i*x) + exp(i*x))/(2*n)))) + exp(i*(2*x + 2*asin(i*(-exp(-i*x) + exp(i*x))/(2*n))))))
Abrimos la expresión [src] 
                                                  sin(x)                                                  
----------------------------------------------------------------------------------------------------------
        _____________                                     _____________                                   
       /        2                   3                    /        2                                       
      /      sin (x)           4*sin (x)*cos(x)         /      sin (x)     2                              
2*   /   1 - ------- *sin(x) - ---------------- - 4*   /   1 - ------- *cos (x)*sin(x) + 2*n*cos(x)*sin(x)
    /            2                    n               /            2                                      
  \/            n                                   \/            n
sin⁡(x)2nsin⁡(x)cos⁡(x)−41−sin⁡2(x)n2sin⁡(x)cos⁡2(x)+21−sin⁡2(x)n2sin⁡(x)−4sin⁡3(x)cos⁡(x)n\frac{\sin{\left(x \right)}}{2 n \sin{\left(x \right)} \cos{\left(x \right)} - 4 \sqrt{1 - \frac{\sin^{2}{\left(x \right)}}{n^{2}}} \sin{\left(x \right)} \cos^{2}{\left(x \right)} + 2 \sqrt{1 - \frac{\sin^{2}{\left(x \right)}}{n^{2}}} \sin{\left(x \right)} - \frac{4 \sin^{3}{\left(x \right)} \cos{\left(x \right)}}{n}} 
sin(x)/(2*sqrt(1 - sin(x)^2/n^2)*sin(x) - 4*sin(x)^3*cos(x)/n - 4*sqrt(1 - sin(x)^2/n^2)*cos(x)^2*sin(x) + 2*n*cos(x)*sin(x))
Parte trigonométrica [src] 
                        /x\    /x   pi\                     
                     cos|-|*cos|- - --|                     
                        \2/    \2   2 /                     
------------------------------------------------------------
     /        /   /    pi\\\    /        /   /    pi\\     \
     |        |cos|x - --|||    |        |cos|x - --||     |
     |        |   \    2 /||    |        |   \    2 /|   pi|
n*cos|x - asin|-----------||*cos|x - asin|-----------| - --|
     \        \     n     //    \        \     n     /   2 /
cos⁡(x2)cos⁡(x2−π2)ncos⁡(x−asin⁡(cos⁡(x−π2)n))cos⁡(x−asin⁡(cos⁡(x−π2)n)−π2)\frac{\cos{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{n \cos{\left(x - \operatorname{asin}{\left(\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{n} \right)} \right)} \cos{\left(x - \operatorname{asin}{\left(\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{n} \right)} - \frac{\pi}{2} \right)}} 
/          2/        /   1    \\  \                             
|       sec |x - asin|--------||  |                             
|           \        \n*csc(x)//  |    /        /   1    \   pi\
|1 + -----------------------------|*sec|x - asin|--------| - --|
|       2/        /   1    \   pi\|    \        \n*csc(x)/   2 /
|    sec |x - asin|--------| - --||                             
\        \        \n*csc(x)/   2 //                             
----------------------------------------------------------------
               /x\    /        /   1    \\    /x   pi\          
          n*sec|-|*sec|x - asin|--------||*sec|- - --|          
               \2/    \        \n*csc(x)//    \2   2 /
(sec⁡2(x−asin⁡(1ncsc⁡(x)))sec⁡2(x−asin⁡(1ncsc⁡(x))−π2)+1)sec⁡(x−asin⁡(1ncsc⁡(x))−π2)nsec⁡(x2)sec⁡(x2−π2)sec⁡(x−asin⁡(1ncsc⁡(x)))\frac{\left(\frac{\sec^{2}{\left(x - \operatorname{asin}{\left(\frac{1}{n \csc{\left(x \right)}} \right)} \right)}}{\sec^{2}{\left(x - \operatorname{asin}{\left(\frac{1}{n \csc{\left(x \right)}} \right)} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(x - \operatorname{asin}{\left(\frac{1}{n \csc{\left(x \right)}} \right)} - \frac{\pi}{2} \right)}}{n \sec{\left(\frac{x}{2} \right)} \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)} \sec{\left(x - \operatorname{asin}{\left(\frac{1}{n \csc{\left(x \right)}} \right)} \right)}} 
                   /x\    /x\                
                cos|-|*sin|-|                
                   \2/    \2/                
---------------------------------------------
     /        /sin(x)\\    /        /sin(x)\\
n*cos|x - asin|------||*sin|x - asin|------||
     \        \  n   //    \        \  n   //
sin⁡(x2)cos⁡(x2)nsin⁡(x−asin⁡(sin⁡(x)n))cos⁡(x−asin⁡(sin⁡(x)n))\frac{\sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}}{n \sin{\left(x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)} \right)} \cos{\left(x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)} \right)}} 
              sin(x)              
----------------------------------
     /        /sin(x)\         pi\
n*cos|- 2*asin|------| + 2*x - --|
     \        \  n   /         2 /
sin⁡(x)ncos⁡(2x−2asin⁡(sin⁡(x)n)−π2)\frac{\sin{\left(x \right)}}{n \cos{\left(2 x - 2 \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)} - \frac{\pi}{2} \right)}} 
               /        /        /         /x\   \\\       
             2 |        |        |    2*tan|-|   |||       
/       2/x\\  |       2|        |         \2/   |||    /x\
|1 - tan |-|| *|1 + tan |x - asin|---------------|||*tan|-|
\        \4//  |        |        |  /       2/x\\|||    \2/
               |        |        |n*|1 + tan |-|||||       
               \        \        \  \        \2/////       
-----------------------------------------------------------
                          /        /         /x\   \\      
                     2    |        |    2*tan|-|   ||      
        /       2/x\\     |        |         \2/   ||      
      n*|1 + tan |-|| *tan|x - asin|---------------||      
        \        \4//     |        |  /       2/x\\||      
                          |        |n*|1 + tan |-||||      
                          \        \  \        \2////
(1−tan⁡2(x4))2(tan⁡2(x−asin⁡(2tan⁡(x2)n(tan⁡2(x2)+1)))+1)tan⁡(x2)n(tan⁡2(x4)+1)2tan⁡(x−asin⁡(2tan⁡(x2)n(tan⁡2(x2)+1)))\frac{\left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{2} \left(\tan^{2}{\left(x - \operatorname{asin}{\left(\frac{2 \tan{\left(\frac{x}{2} \right)}}{n \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} \right)} \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}}{n \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \tan{\left(x - \operatorname{asin}{\left(\frac{2 \tan{\left(\frac{x}{2} \right)}}{n \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} \right)} \right)}} 
                                     2                                       
/        /        /         /x\   \\\                                        
|        |        |    2*tan|-|   |||                                        
|        |        |         \2/   |||                                        
|        |    asin|---------------|||                                        
|        |        |  /       2/x\\|||             /        /         /x\   \\
|        |        |n*|1 + tan |-|||||             |        |    2*tan|-|   ||
|       2|x       \  \        \2///||     2/x\    |        |         \2/   ||
|1 + tan |- - ---------------------|| *tan |-|*tan|x - asin|---------------||
\        \2             2          //      \4/    |        |  /       2/x\\||
                                                  |        |n*|1 + tan |-||||
                                                  \        \  \        \2////
-----------------------------------------------------------------------------
                                       /        /         /x\   \\           
                                       |        |    2*tan|-|   ||           
                                       |        |         \2/   ||           
                                       |    asin|---------------||           
                                       |        |  /       2/x\\||           
                          2            |        |n*|1 + tan |-||||           
             /       2/x\\     /x\    2|x       \  \        \2///|           
           n*|1 + tan |-|| *tan|-|*tan |- - ---------------------|           
             \        \4//     \2/     \2             2          /
(tan⁡2(x2−asin⁡(2tan⁡(x2)n(tan⁡2(x2)+1))2)+1)2tan⁡2(x4)tan⁡(x−asin⁡(2tan⁡(x2)n(tan⁡2(x2)+1)))n(tan⁡2(x4)+1)2tan⁡(x2)tan⁡2(x2−asin⁡(2tan⁡(x2)n(tan⁡2(x2)+1))2)\frac{\left(\tan^{2}{\left(\frac{x}{2} - \frac{\operatorname{asin}{\left(\frac{2 \tan{\left(\frac{x}{2} \right)}}{n \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} \right)}}{2} \right)} + 1\right)^{2} \tan^{2}{\left(\frac{x}{4} \right)} \tan{\left(x - \operatorname{asin}{\left(\frac{2 \tan{\left(\frac{x}{2} \right)}}{n \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} \right)} \right)}}{n \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \tan{\left(\frac{x}{2} \right)} \tan^{2}{\left(\frac{x}{2} - \frac{\operatorname{asin}{\left(\frac{2 \tan{\left(\frac{x}{2} \right)}}{n \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} \right)}}{2} \right)}} 
   /        /   1    \      \
csc|- 2*asin|--------| + 2*x|
   \        \n*csc(x)/      /
-----------------------------
           n*csc(x)
csc⁡(2x−2asin⁡(1ncsc⁡(x)))ncsc⁡(x)\frac{\csc{\left(2 x - 2 \operatorname{asin}{\left(\frac{1}{n \csc{\left(x \right)}} \right)} \right)}}{n \csc{\left(x \right)}} 
           sin(x)          
---------------------------
     /  /        /sin(x)\\\
n*sin|2*|x - asin|------|||
     \  \        \  n   ///
sin⁡(x)nsin⁡(2(x−asin⁡(sin⁡(x)n)))\frac{\sin{\left(x \right)}}{n \sin{\left(2 \left(x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)}\right) \right)}} 
   /pi               /      1      \\
sec|-- - 2*x + 2*asin|-------------||
   |2                |     /pi    \||
   |                 |n*sec|-- - x|||
   \                 \     \2     ///
-------------------------------------
                 /    pi\            
            n*sec|x - --|            
                 \    2 /
sec⁡(−2x+2asin⁡(1nsec⁡(−x+π2))+π2)nsec⁡(x−π2)\frac{\sec{\left(- 2 x + 2 \operatorname{asin}{\left(\frac{1}{n \sec{\left(- x + \frac{\pi}{2} \right)}} \right)} + \frac{\pi}{2} \right)}}{n \sec{\left(x - \frac{\pi}{2} \right)}} 
   /        /      1      \         pi\
sec|- 2*asin|-------------| + 2*x - --|
   |        |     /    pi\|         2 |
   |        |n*sec|x - --||           |
   \        \     \    2 //           /
---------------------------------------
                  /    pi\             
             n*sec|x - --|             
                  \    2 /
sec⁡(2x−2asin⁡(1nsec⁡(x−π2))−π2)nsec⁡(x−π2)\frac{\sec{\left(2 x - 2 \operatorname{asin}{\left(\frac{1}{n \sec{\left(x - \frac{\pi}{2} \right)}} \right)} - \frac{\pi}{2} \right)}}{n \sec{\left(x - \frac{\pi}{2} \right)}} 
   /x\    /        /   1    \\    /        /sin(x)\\    /x\
cos|-|*csc|x - asin|--------||*sec|x - asin|------||*sin|-|
   \2/    \        \n*csc(x)//    \        \  n   //    \2/
-----------------------------------------------------------
                             n
sin⁡(x2)cos⁡(x2)csc⁡(x−asin⁡(1ncsc⁡(x)))sec⁡(x−asin⁡(sin⁡(x)n))n\frac{\sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} \csc{\left(x - \operatorname{asin}{\left(\frac{1}{n \csc{\left(x \right)}} \right)} \right)} \sec{\left(x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)} \right)}}{n} 
/       2/pi           /   1    \\\                        
|    csc |-- - x + asin|--------|||                        
|        \2            \n*csc(x)//|    /        /   1    \\
|1 + -----------------------------|*csc|x - asin|--------||
|          2/        /   1    \\  |    \        \n*csc(x)//
|       csc |x - asin|--------||  |                        
\           \        \n*csc(x)//  /                        
-----------------------------------------------------------
          /x\    /pi   x\    /pi           /   1    \\     
     n*csc|-|*csc|-- - -|*csc|-- - x + asin|--------||     
          \2/    \2    2/    \2            \n*csc(x)//
(1+csc⁡2(−x+asin⁡(1ncsc⁡(x))+π2)csc⁡2(x−asin⁡(1ncsc⁡(x))))csc⁡(x−asin⁡(1ncsc⁡(x)))ncsc⁡(x2)csc⁡(−x2+π2)csc⁡(−x+asin⁡(1ncsc⁡(x))+π2)\frac{\left(1 + \frac{\csc^{2}{\left(- x + \operatorname{asin}{\left(\frac{1}{n \csc{\left(x \right)}} \right)} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x - \operatorname{asin}{\left(\frac{1}{n \csc{\left(x \right)}} \right)} \right)}}\right) \csc{\left(x - \operatorname{asin}{\left(\frac{1}{n \csc{\left(x \right)}} \right)} \right)}}{n \csc{\left(\frac{x}{2} \right)} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)} \csc{\left(- x + \operatorname{asin}{\left(\frac{1}{n \csc{\left(x \right)}} \right)} + \frac{\pi}{2} \right)}} 
                                                         /        /         /x\   \\
              2                                          |        |    2*cot|-|   ||
/        2/x\\  /                   1               \    |        |         \2/   ||
|-1 + cot |-|| *|1 + -------------------------------|*cot|x - asin|---------------||
\         \4//  |        /        /         /x\   \\|    |        |  /       2/x\\||
                |        |        |    2*cot|-|   |||    |        |n*|1 + cot |-||||
                |       2|        |         \2/   |||    \        \  \        \2////
                |    cot |x - asin|---------------|||                               
                |        |        |  /       2/x\\|||                               
                |        |        |n*|1 + cot |-|||||                               
                \        \        \  \        \2/////                               
------------------------------------------------------------------------------------
                                             2                                      
                                /       2/x\\     /x\                               
                              n*|1 + cot |-|| *cot|-|                               
                                \        \4//     \2/
(1+1cot⁡2(x−asin⁡(2cot⁡(x2)n(cot⁡2(x2)+1))))(cot⁡2(x4)−1)2cot⁡(x−asin⁡(2cot⁡(x2)n(cot⁡2(x2)+1)))n(cot⁡2(x4)+1)2cot⁡(x2)\frac{\left(1 + \frac{1}{\cot^{2}{\left(x - \operatorname{asin}{\left(\frac{2 \cot{\left(\frac{x}{2} \right)}}{n \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} \right)} \right)}}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{2} \cot{\left(x - \operatorname{asin}{\left(\frac{2 \cot{\left(\frac{x}{2} \right)}}{n \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} \right)} \right)}}{n \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \cot{\left(\frac{x}{2} \right)}} 
   2/x\    /x\    /        /sin(x)\\
sin |-|*cot|-|*tan|x - asin|------||
    \2/    \2/    \        \  n   //
------------------------------------
           2/        /sin(x)\\      
      n*sin |x - asin|------||      
            \        \  n   //
sin⁡2(x2)tan⁡(x−asin⁡(sin⁡(x)n))cot⁡(x2)nsin⁡2(x−asin⁡(sin⁡(x)n))\frac{\sin^{2}{\left(\frac{x}{2} \right)} \tan{\left(x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)} \right)} \cot{\left(\frac{x}{2} \right)}}{n \sin^{2}{\left(x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)} \right)}} 
              /    pi\            
           cos|x - --|            
              \    2 /            
----------------------------------
     /        /sin(x)\         pi\
n*cos|- 2*asin|------| + 2*x - --|
     \        \  n   /         2 /
cos⁡(x−π2)ncos⁡(2x−2asin⁡(sin⁡(x)n)−π2)\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{n \cos{\left(2 x - 2 \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)} - \frac{\pi}{2} \right)}} 
                                                  2                            
             /        /        /         /x\   \\\                             
             |        |        |    2*cot|-|   |||                             
             |        |        |         \2/   |||                             
             |        |    asin|---------------|||                             
             |        |        |  /       2/x\\|||                             
             |        |        |n*|1 + cot |-|||||                             
             |       2|x       \  \        \2///||     2/x\    /x\             
             |1 + cot |- - ---------------------|| *cot |-|*cot|-|             
             \        \2             2          //      \4/    \2/             
-------------------------------------------------------------------------------
                                                    /        /         /x\   \\
                                                    |        |    2*cot|-|   ||
                                                    |        |         \2/   ||
                                                    |    asin|---------------||
                    /        /         /x\   \\     |        |  /       2/x\\||
               2    |        |    2*cot|-|   ||     |        |n*|1 + cot |-||||
  /       2/x\\     |        |         \2/   ||    2|x       \  \        \2///|
n*|1 + cot |-|| *cot|x - asin|---------------||*cot |- - ---------------------|
  \        \4//     |        |  /       2/x\\||     \2             2          /
                    |        |n*|1 + cot |-||||                                
                    \        \  \        \2////
(cot⁡2(x2−asin⁡(2cot⁡(x2)n(cot⁡2(x2)+1))2)+1)2cot⁡2(x4)cot⁡(x2)n(cot⁡2(x4)+1)2cot⁡2(x2−asin⁡(2cot⁡(x2)n(cot⁡2(x2)+1))2)cot⁡(x−asin⁡(2cot⁡(x2)n(cot⁡2(x2)+1)))\frac{\left(\cot^{2}{\left(\frac{x}{2} - \frac{\operatorname{asin}{\left(\frac{2 \cot{\left(\frac{x}{2} \right)}}{n \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} \right)}}{2} \right)} + 1\right)^{2} \cot^{2}{\left(\frac{x}{4} \right)} \cot{\left(\frac{x}{2} \right)}}{n \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \cot^{2}{\left(\frac{x}{2} - \frac{\operatorname{asin}{\left(\frac{2 \cot{\left(\frac{x}{2} \right)}}{n \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} \right)}}{2} \right)} \cot{\left(x - \operatorname{asin}{\left(\frac{2 \cot{\left(\frac{x}{2} \right)}}{n \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} \right)} \right)}} 
           4/x\    2/x\    /        /sin(x)\\         
        cos |-|*tan |-|*tan|x - asin|------||         
            \4/     \4/    \        \  n   //         
------------------------------------------------------
      /        /sin(x)\\            /        /sin(x)\\
      |x - asin|------||            |x - asin|------||
     4|        \  n   /|    /x\    2|        \  n   /|
n*cos |----------------|*tan|-|*tan |----------------|
      \       2        /    \2/     \       2        /
cos⁡4(x4)tan⁡2(x4)tan⁡(x−asin⁡(sin⁡(x)n))ncos⁡4(x−asin⁡(sin⁡(x)n)2)tan⁡(x2)tan⁡2(x−asin⁡(sin⁡(x)n)2)\frac{\cos^{4}{\left(\frac{x}{4} \right)} \tan^{2}{\left(\frac{x}{4} \right)} \tan{\left(x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)} \right)}}{n \cos^{4}{\left(\frac{x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)}}{2} \right)} \tan{\left(\frac{x}{2} \right)} \tan^{2}{\left(\frac{x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)}}{2} \right)}} 
   2/        /      1      \   pi\    /        /   1    \\
sec |x - asin|-------------| - --|*sec|x - asin|--------||
    |        |     /    pi\|   2 |    \        \n*csc(x)//
    |        |n*sec|x - --||     |                        
    \        \     \    2 //     /                        
----------------------------------------------------------
         /x\    /x   pi\    /        /   1    \   pi\     
    n*sec|-|*sec|- - --|*sec|x - asin|--------| - --|     
         \2/    \2   2 /    \        \n*csc(x)/   2 /
sec⁡(x−asin⁡(1ncsc⁡(x)))sec⁡2(x−asin⁡(1nsec⁡(x−π2))−π2)nsec⁡(x2)sec⁡(x2−π2)sec⁡(x−asin⁡(1ncsc⁡(x))−π2)\frac{\sec{\left(x - \operatorname{asin}{\left(\frac{1}{n \csc{\left(x \right)}} \right)} \right)} \sec^{2}{\left(x - \operatorname{asin}{\left(\frac{1}{n \sec{\left(x - \frac{\pi}{2} \right)}} \right)} - \frac{\pi}{2} \right)}}{n \sec{\left(\frac{x}{2} \right)} \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)} \sec{\left(x - \operatorname{asin}{\left(\frac{1}{n \csc{\left(x \right)}} \right)} - \frac{\pi}{2} \right)}} 
   2/x\ /       2/        /sin(x)\\\    /x\
cos |-|*|1 + tan |x - asin|------|||*tan|-|
    \2/ \        \        \  n   ///    \2/
-------------------------------------------
               /        /sin(x)\\          
          n*tan|x - asin|------||          
               \        \  n   //
(tan⁡2(x−asin⁡(sin⁡(x)n))+1)cos⁡2(x2)tan⁡(x2)ntan⁡(x−asin⁡(sin⁡(x)n))\frac{\left(\tan^{2}{\left(x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)} \right)} + 1\right) \cos^{2}{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} \right)}}{n \tan{\left(x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)} \right)}} 
   /pi               /      1      \\
sec|-- - 2*x + 2*asin|-------------||
   |2                |     /    pi\||
   |                 |n*sec|x - --|||
   \                 \     \    2 ///
-------------------------------------
                 /    pi\            
            n*sec|x - --|            
                 \    2 /
sec⁡(−2x+2asin⁡(1nsec⁡(x−π2))+π2)nsec⁡(x−π2)\frac{\sec{\left(- 2 x + 2 \operatorname{asin}{\left(\frac{1}{n \sec{\left(x - \frac{\pi}{2} \right)}} \right)} + \frac{\pi}{2} \right)}}{n \sec{\left(x - \frac{\pi}{2} \right)}} 
                               n*sin(2*x)                               
------------------------------------------------------------------------
                                          2                             
  /                  _____________       \                              
  |                 /        2           |                              
  |   2            /      sin (x)        |            /        /sin(x)\\
4*|sin (x) + n*   /   1 - ------- *cos(x)| *cos(x)*tan|x - asin|------||
  |              /            2          |            \        \  n   //
  \            \/            n           /
nsin⁡(2x)4(n1−sin⁡2(x)n2cos⁡(x)+sin⁡2(x))2cos⁡(x)tan⁡(x−asin⁡(sin⁡(x)n))\frac{n \sin{\left(2 x \right)}}{4 \left(n \sqrt{1 - \frac{\sin^{2}{\left(x \right)}}{n^{2}}} \cos{\left(x \right)} + \sin^{2}{\left(x \right)}\right)^{2} \cos{\left(x \right)} \tan{\left(x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)} \right)}} 
                     /           4/        /sin(x)\\  \                            
                     |      4*sin |x - asin|------||  |                            
   2/x\    2/pi   x\ |            \        \  n   //  |    /        /sin(x)\      \
sin |-|*sin |-- + -|*|1 + ----------------------------|*sin|- 2*asin|------| + 2*x|
    \2/     \2    2/ |       2/        /sin(x)\      \|    \        \  n   /      /
                     |    sin |- 2*asin|------| + 2*x||                            
                     \        \        \  n   /      //                            
-----------------------------------------------------------------------------------
                                      2/        /sin(x)\\                          
                          n*sin(x)*sin |x - asin|------||                          
                                       \        \  n   //
(4sin⁡4(x−asin⁡(sin⁡(x)n))sin⁡2(2x−2asin⁡(sin⁡(x)n))+1)sin⁡2(x2)sin⁡2(x2+π2)sin⁡(2x−2asin⁡(sin⁡(x)n))nsin⁡(x)sin⁡2(x−asin⁡(sin⁡(x)n))\frac{\left(\frac{4 \sin^{4}{\left(x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)} \right)}}{\sin^{2}{\left(2 x - 2 \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)} \right)}} + 1\right) \sin^{2}{\left(\frac{x}{2} \right)} \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{2} \right)} \sin{\left(2 x - 2 \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)} \right)}}{n \sin{\left(x \right)} \sin^{2}{\left(x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)} \right)}} 
                /          4/        /sin(x)\\ \                          
                |     4*sin |x - asin|------|| |                          
   2/x\    2/x\ |           \        \  n   // |    /  /        /sin(x)\\\
cos |-|*sin |-|*|1 + --------------------------|*sin|2*|x - asin|------|||
    \2/     \2/ |       2/  /        /sin(x)\\\|    \  \        \  n   ///
                |    sin |2*|x - asin|------||||                          
                \        \  \        \  n   ////                          
--------------------------------------------------------------------------
                                 2/        /sin(x)\\                      
                     n*sin(x)*sin |x - asin|------||                      
                                  \        \  n   //
(4sin⁡4(x−asin⁡(sin⁡(x)n))sin⁡2(2(x−asin⁡(sin⁡(x)n)))+1)sin⁡2(x2)sin⁡(2(x−asin⁡(sin⁡(x)n)))cos⁡2(x2)nsin⁡(x)sin⁡2(x−asin⁡(sin⁡(x)n))\frac{\left(\frac{4 \sin^{4}{\left(x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)} \right)}}{\sin^{2}{\left(2 \left(x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)}\right) \right)}} + 1\right) \sin^{2}{\left(\frac{x}{2} \right)} \sin{\left(2 \left(x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)}\right) \right)} \cos^{2}{\left(\frac{x}{2} \right)}}{n \sin{\left(x \right)} \sin^{2}{\left(x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)} \right)}} 
   /x\    /        /sin(x)\\    /        /      1      \\    /x\
cos|-|*csc|x - asin|------||*sec|x - asin|-------------||*sin|-|
   \2/    \        \  n   //    |        |     /    pi\||    \2/
                                |        |n*sec|x - --|||       
                                \        \     \    2 ///       
----------------------------------------------------------------
                               n
sin⁡(x2)cos⁡(x2)csc⁡(x−asin⁡(sin⁡(x)n))sec⁡(x−asin⁡(1nsec⁡(x−π2)))n\frac{\sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} \csc{\left(x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)} \right)} \sec{\left(x - \operatorname{asin}{\left(\frac{1}{n \sec{\left(x - \frac{\pi}{2} \right)}} \right)} \right)}}{n} 
   /        /   1    \      \       
csc|- 2*asin|--------| + 2*x|*sin(x)
   \        \n*csc(x)/      /       
------------------------------------
                 n
sin⁡(x)csc⁡(2x−2asin⁡(1ncsc⁡(x)))n\frac{\sin{\left(x \right)} \csc{\left(2 x - 2 \operatorname{asin}{\left(\frac{1}{n \csc{\left(x \right)}} \right)} \right)}}{n} 
   /  /        /sin(x)\\\       
csc|2*|x - asin|------|||*sin(x)
   \  \        \  n   ///       
--------------------------------
               n
sin⁡(x)csc⁡(2(x−asin⁡(sin⁡(x)n)))n\frac{\sin{\left(x \right)} \csc{\left(2 \left(x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)}\right) \right)}}{n} 
 /        /        /         /x\   \\\        
 |        |        |    2*tan|-|   |||        
 |       2|        |         \2/   |||    /x\ 
 |1 + tan |x - asin|---------------|||*tan|-| 
 |        |        |  /       2/x\\|||    \2/ 
 |        |        |n*|1 + tan |-|||||        
 \        \        \  \        \2/////        
----------------------------------------------
                   /        /         /x\   \\
                   |        |    2*tan|-|   ||
  /       2/x\\    |        |         \2/   ||
n*|1 + tan |-||*tan|x - asin|---------------||
  \        \2//    |        |  /       2/x\\||
                   |        |n*|1 + tan |-||||
                   \        \  \        \2////
(tan⁡2(x−asin⁡(2tan⁡(x2)n(tan⁡2(x2)+1)))+1)tan⁡(x2)n(tan⁡2(x2)+1)tan⁡(x−asin⁡(2tan⁡(x2)n(tan⁡2(x2)+1)))\frac{\left(\tan^{2}{\left(x - \operatorname{asin}{\left(\frac{2 \tan{\left(\frac{x}{2} \right)}}{n \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} \right)} \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}}{n \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \tan{\left(x - \operatorname{asin}{\left(\frac{2 \tan{\left(\frac{x}{2} \right)}}{n \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} \right)} \right)}} 
   /        /   1    \\    /pi           /   1    \\
csc|x - asin|--------||*csc|-- - x + asin|--------||
   \        \n*csc(x)//    \2            \n*csc(x)//
----------------------------------------------------
                     /x\    /pi   x\                
                n*csc|-|*csc|-- - -|                
                     \2/    \2    2/
csc⁡(x−asin⁡(1ncsc⁡(x)))csc⁡(−x+asin⁡(1ncsc⁡(x))+π2)ncsc⁡(x2)csc⁡(−x2+π2)\frac{\csc{\left(x - \operatorname{asin}{\left(\frac{1}{n \csc{\left(x \right)}} \right)} \right)} \csc{\left(- x + \operatorname{asin}{\left(\frac{1}{n \csc{\left(x \right)}} \right)} + \frac{\pi}{2} \right)}}{n \csc{\left(\frac{x}{2} \right)} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} 
 /        /        /         /x\   \\\        
 |        |        |    2*cot|-|   |||        
 |       2|        |         \2/   |||    /x\ 
 |1 + cot |x - asin|---------------|||*cot|-| 
 |        |        |  /       2/x\\|||    \2/ 
 |        |        |n*|1 + cot |-|||||        
 \        \        \  \        \2/////        
----------------------------------------------
                   /        /         /x\   \\
                   |        |    2*cot|-|   ||
  /       2/x\\    |        |         \2/   ||
n*|1 + cot |-||*cot|x - asin|---------------||
  \        \2//    |        |  /       2/x\\||
                   |        |n*|1 + cot |-||||
                   \        \  \        \2////
(cot⁡2(x−asin⁡(2cot⁡(x2)n(cot⁡2(x2)+1)))+1)cot⁡(x2)n(cot⁡2(x2)+1)cot⁡(x−asin⁡(2cot⁡(x2)n(cot⁡2(x2)+1)))\frac{\left(\cot^{2}{\left(x - \operatorname{asin}{\left(\frac{2 \cot{\left(\frac{x}{2} \right)}}{n \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} \right)} \right)} + 1\right) \cot{\left(\frac{x}{2} \right)}}{n \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \cot{\left(x - \operatorname{asin}{\left(\frac{2 \cot{\left(\frac{x}{2} \right)}}{n \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} \right)} \right)}} 
             2                                            
/       2/x\\     4/x\ /       2/        /sin(x)\\\    /x\
|1 - tan |-|| *cos |-|*|1 + tan |x - asin|------|||*tan|-|
\        \4//      \4/ \        \        \  n   ///    \2/
----------------------------------------------------------
                      /        /sin(x)\\                  
                 n*tan|x - asin|------||                  
                      \        \  n   //
(1−tan⁡2(x4))2(tan⁡2(x−asin⁡(sin⁡(x)n))+1)cos⁡4(x4)tan⁡(x2)ntan⁡(x−asin⁡(sin⁡(x)n))\frac{\left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{2} \left(\tan^{2}{\left(x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)} \right)} + 1\right) \cos^{4}{\left(\frac{x}{4} \right)} \tan{\left(\frac{x}{2} \right)}}{n \tan{\left(x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)} \right)}} 
                 /    pi\              
              cos|x - --|              
                 \    2 /              
---------------------------------------
     /        /   /    pi\\           \
     |        |cos|x - --||           |
     |        |   \    2 /|         pi|
n*cos|- 2*asin|-----------| + 2*x - --|
     \        \     n     /         2 /
cos⁡(x−π2)ncos⁡(2x−2asin⁡(cos⁡(x−π2)n)−π2)\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{n \cos{\left(2 x - 2 \operatorname{asin}{\left(\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{n} \right)} - \frac{\pi}{2} \right)}} 
              2                                                                  
/        2/x\\     4/x\ /       2/        /sin(x)\\\    /        /sin(x)\\    /x\
|-1 + cot |-|| *sin |-|*|1 + tan |x - asin|------|||*cot|x - asin|------||*tan|-|
\         \4//      \4/ \        \        \  n   ///    \        \  n   //    \2/
---------------------------------------------------------------------------------
                                        n
(tan⁡2(x−asin⁡(sin⁡(x)n))+1)(cot⁡2(x4)−1)2sin⁡4(x4)tan⁡(x2)cot⁡(x−asin⁡(sin⁡(x)n))n\frac{\left(\tan^{2}{\left(x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)} \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{x}{4} \right)} \tan{\left(\frac{x}{2} \right)} \cot{\left(x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)} \right)}}{n} 
/        /        /   /    pi\\     \\                                              
|        |        |cos|x - --||     ||                                              
|       2|        |   \    2 /|   pi||           /        /   /    pi\\\            
|    cos |x - asin|-----------| - --||           |        |cos|x - --|||            
|        \        \     n     /   2 /|    /x\    |        |   \    2 /||    /x   pi\
|1 + --------------------------------|*cos|-|*cos|x - asin|-----------||*cos|- - --|
|          /        /   /    pi\\\   |    \2/    \        \     n     //    \2   2 /
|          |        |cos|x - --|||   |                                              
|         2|        |   \    2 /||   |                                              
|      cos |x - asin|-----------||   |                                              
\          \        \     n     //   /                                              
------------------------------------------------------------------------------------
                              /        /   /    pi\\     \                          
                              |        |cos|x - --||     |                          
                              |        |   \    2 /|   pi|                          
                         n*cos|x - asin|-----------| - --|                          
                              \        \     n     /   2 /
(1+cos⁡2(x−asin⁡(cos⁡(x−π2)n)−π2)cos⁡2(x−asin⁡(cos⁡(x−π2)n)))cos⁡(x2)cos⁡(x2−π2)cos⁡(x−asin⁡(cos⁡(x−π2)n))ncos⁡(x−asin⁡(cos⁡(x−π2)n)−π2)\frac{\left(1 + \frac{\cos^{2}{\left(x - \operatorname{asin}{\left(\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{n} \right)} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x - \operatorname{asin}{\left(\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{n} \right)} \right)}}\right) \cos{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)} \cos{\left(x - \operatorname{asin}{\left(\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{n} \right)} \right)}}{n \cos{\left(x - \operatorname{asin}{\left(\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{n} \right)} - \frac{\pi}{2} \right)}} 
     2/x\    4/x\    /x\    /        /sin(x)\\ 
  cot |-|*sin |-|*cot|-|*tan|x - asin|------|| 
      \4/     \4/    \2/    \        \  n   // 
-----------------------------------------------
      /        /sin(x)\\     /        /sin(x)\\
      |x - asin|------||     |x - asin|------||
     2|        \  n   /|    4|        \  n   /|
n*cot |----------------|*sin |----------------|
      \       2        /     \       2        /
sin⁡4(x4)tan⁡(x−asin⁡(sin⁡(x)n))cot⁡2(x4)cot⁡(x2)nsin⁡4(x−asin⁡(sin⁡(x)n)2)cot⁡2(x−asin⁡(sin⁡(x)n)2)\frac{\sin^{4}{\left(\frac{x}{4} \right)} \tan{\left(x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)} \right)} \cot^{2}{\left(\frac{x}{4} \right)} \cot{\left(\frac{x}{2} \right)}}{n \sin^{4}{\left(\frac{x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)}}{2} \right)} \cot^{2}{\left(\frac{x - \operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{n} \right)}}{2} \right)}} 
cot(x/4)^2*sin(x/4)^4*cot(x/2)*tan(x - asin(sin(x)/n))/(n*cot((x - asin(sin(x)/n))/2)^2*sin((x - asin(sin(x)/n))/2)^4)
    © Sr Examen – Калькулятор