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Simplificación de expresiones/ Descomposición en fracciones simples/ sin(x)/(2*sqrt(1-cos(x)^2)*sqrt(acos(cos(x))))

¿Cómo vas a descomponer esta sin(x)/(2*sqrt(1-cos(x)^2)*sqrt(acos(cos(x)))) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
               sin(x)              
-----------------------------------
     _____________                 
    /        2       ______________
2*\/  1 - cos (x) *\/ acos(cos(x))
sin⁡(x)21−cos⁡2(x)acos⁡(cos⁡(x))\frac{\sin{\left(x \right)}}{2 \sqrt{1 - \cos^{2}{\left(x \right)}} \sqrt{\operatorname{acos}{\left(\cos{\left(x \right)} \right)}}} 
sin(x)/(((2*sqrt(1 - cos(x)^2))*sqrt(acos(cos(x)))))
Simplificación general [src] 
             sin(x)            
-------------------------------
     _________                 
    /    2       ______________
2*\/  sin (x) *\/ acos(cos(x))
sin⁡(x)2sin⁡2(x)acos⁡(cos⁡(x))\frac{\sin{\left(x \right)}}{2 \sqrt{\sin^{2}{\left(x \right)}} \sqrt{\operatorname{acos}{\left(\cos{\left(x \right)} \right)}}} 
sin(x)/(2*sqrt(sin(x)^2)*sqrt(acos(cos(x))))
Respuesta numérica [src] 
0.5*(1.0 - cos(x)^2)^(-0.5)*acos(cos(x))^(-0.5)*sin(x)
0.5*(1.0 - cos(x)^2)^(-0.5)*acos(cos(x))^(-0.5)*sin(x)
Potencias [src] 
                     /   -I*x    I*x\                  
                  -I*\- e     + e   /                  
-------------------------------------------------------
        _____________________                          
       /                   2       ____________________
      /      / I*x    -I*x\       /     / I*x    -I*x\ 
     /       |e      e    |      /      |e      e    | 
4*  /    1 - |---- + -----|  *  /   acos|---- + -----| 
  \/         \ 2       2  /   \/        \ 2       2  /
−i(eix−e−ix)41−(eix2+e−ix2)2acos⁡(eix2+e−ix2)- \frac{i \left(e^{i x} - e^{- i x}\right)}{4 \sqrt{1 - \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2}} \sqrt{\operatorname{acos}{\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2} \right)}}} 
              sin(x)             
---------------------------------
     ____________________________
    / /       2   \              
2*\/  \1 - cos (x)/*acos(cos(x))
sin⁡(x)2(1−cos⁡2(x))acos⁡(cos⁡(x))\frac{\sin{\left(x \right)}}{2 \sqrt{\left(1 - \cos^{2}{\left(x \right)}\right) \operatorname{acos}{\left(\cos{\left(x \right)} \right)}}} 
sin(x)/(2*sqrt((1 - cos(x)^2)*acos(cos(x))))
Denominador racional [src] 
        _____________            
       /        2                
    -\/  1 - cos (x) *sin(x)     
---------------------------------
  /        2   \   ______________
2*\-1 + cos (x)/*\/ acos(cos(x))
−1−cos⁡2(x)sin⁡(x)2(cos⁡2(x)−1)acos⁡(cos⁡(x))- \frac{\sqrt{1 - \cos^{2}{\left(x \right)}} \sin{\left(x \right)}}{2 \left(\cos^{2}{\left(x \right)} - 1\right) \sqrt{\operatorname{acos}{\left(\cos{\left(x \right)} \right)}}} 
-sqrt(1 - cos(x)^2)*sin(x)/(2*(-1 + cos(x)^2)*sqrt(acos(cos(x))))
Combinatoria [src] 
                      sin(x)                      
--------------------------------------------------
    _____________________________   ______________
2*\/ -(1 + cos(x))*(-1 + cos(x)) *\/ acos(cos(x))
sin⁡(x)2−(cos⁡(x)−1)(cos⁡(x)+1)acos⁡(cos⁡(x))\frac{\sin{\left(x \right)}}{2 \sqrt{- \left(\cos{\left(x \right)} - 1\right) \left(\cos{\left(x \right)} + 1\right)} \sqrt{\operatorname{acos}{\left(\cos{\left(x \right)} \right)}}} 
sin(x)/(2*sqrt(-(1 + cos(x))*(-1 + cos(x)))*sqrt(acos(cos(x))))
Parte trigonométrica [src] 
                                   /x\                                
                                cot|-|                                
                                   \2/                                
----------------------------------------------------------------------
           ________________                       ____________________
          /       2/x\                           /     /        2/x\\ 
         /     cot |-|                          /      |-1 + cot |-|| 
        /          \2/      /       2/x\\      /       |         \2/| 
2*     /    -------------- *|1 + cot |-||*    /    acos|------------| 
      /                  2  \        \2//    /         |       2/x\ | 
     /      /       2/x\\                   /          |1 + cot |-| | 
    /       |1 + cot |-||                 \/           \        \2/ / 
  \/        \        \2//
cot⁡(x2)2cot⁡2(x2)(cot⁡2(x2)+1)2(cot⁡2(x2)+1)acos⁡(cot⁡2(x2)−1cot⁡2(x2)+1)\frac{\cot{\left(\frac{x}{2} \right)}}{2 \sqrt{\frac{\cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \sqrt{\operatorname{acos}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}} 
                                   /x\                               
                                tan|-|                               
                                   \2/                               
---------------------------------------------------------------------
           ________________                       ___________________
          /       2/x\                           /     /       2/x\\ 
         /     tan |-|                          /      |1 - tan |-|| 
        /          \2/      /       2/x\\      /       |        \2/| 
2*     /    -------------- *|1 + tan |-||*    /    acos|-----------| 
      /                  2  \        \2//    /         |       2/x\| 
     /      /       2/x\\                   /          |1 + tan |-|| 
    /       |1 + tan |-||                 \/           \        \2// 
  \/        \        \2//
tan⁡(x2)2tan⁡2(x2)(tan⁡2(x2)+1)2(tan⁡2(x2)+1)acos⁡(1−tan⁡2(x2)tan⁡2(x2)+1)\frac{\tan{\left(\frac{x}{2} \right)}}{2 \sqrt{\frac{\tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \sqrt{\operatorname{acos}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}} 
                              ___    /x\                           
                            \/ 2 *cot|-|                           
                                     \2/                           
-------------------------------------------------------------------
                                               ____________________
                    __________________        /     /        2/x\\ 
                   /             2           /      |-1 + cot |-|| 
/       2/x\\     /      -1 + cot (x)       /       |         \2/| 
|1 + cot |-||*   /   1 - ------------ *    /    acos|------------| 
\        \2//   /               2         /         |       2/x\ | 
              \/         1 + cot (x)     /          |1 + cot |-| | 
                                       \/           \        \2/ /
2cot⁡(x2)−cot⁡2(x)−1cot⁡2(x)+1+1(cot⁡2(x2)+1)acos⁡(cot⁡2(x2)−1cot⁡2(x2)+1)\frac{\sqrt{2} \cot{\left(\frac{x}{2} \right)}}{\sqrt{- \frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} + 1} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \sqrt{\operatorname{acos}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}} 
                      1                       
----------------------------------------------
       _____________     ______________       
      /        1        /     /  1   \        
2*   /  1 - ------- *  /  acos|------| *csc(x)
    /          2     \/       \sec(x)/        
  \/        sec (x)
121−1sec⁡2(x)acos⁡(1sec⁡(x))csc⁡(x)\frac{1}{2 \sqrt{1 - \frac{1}{\sec^{2}{\left(x \right)}}} \sqrt{\operatorname{acos}{\left(\frac{1}{\sec{\left(x \right)}} \right)}} \csc{\left(x \right)}} 
                sin(x)                
--------------------------------------
     _________     ___________________
    /    2        /     /   /    pi\\ 
2*\/  sin (x) *  /  acos|sin|x + --|| 
               \/       \   \    2 //
sin⁡(x)2sin⁡2(x)acos⁡(sin⁡(x+π2))\frac{\sin{\left(x \right)}}{2 \sqrt{\sin^{2}{\left(x \right)}} \sqrt{\operatorname{acos}{\left(\sin{\left(x + \frac{\pi}{2} \right)} \right)}}} 
                         ___                       
                       \/ 2                        
---------------------------------------------------
      ______________     ______________            
     /        1         /     /  1   \     /    pi\
2*  /  1 - -------- *  /  acos|------| *sec|x - --|
  \/       sec(2*x)  \/       \sec(x)/     \    2 /
221−1sec⁡(2x)acos⁡(1sec⁡(x))sec⁡(x−π2)\frac{\sqrt{2}}{2 \sqrt{1 - \frac{1}{\sec{\left(2 x \right)}}} \sqrt{\operatorname{acos}{\left(\frac{1}{\sec{\left(x \right)}} \right)}} \sec{\left(x - \frac{\pi}{2} \right)}} 
                             ___    /x\                          
                           \/ 2 *tan|-|                          
                                    \2/                          
-----------------------------------------------------------------
                                              ___________________
                    _________________        /     /       2/x\\ 
                   /            2           /      |1 - tan |-|| 
/       2/x\\     /      1 - tan (x)       /       |        \2/| 
|1 + tan |-||*   /   1 - ----------- *    /    acos|-----------| 
\        \2//   /               2        /         |       2/x\| 
              \/         1 + tan (x)    /          |1 + tan |-|| 
                                      \/           \        \2//
2tan⁡(x2)−1−tan⁡2(x)tan⁡2(x)+1+1(tan⁡2(x2)+1)acos⁡(1−tan⁡2(x2)tan⁡2(x2)+1)\frac{\sqrt{2} \tan{\left(\frac{x}{2} \right)}}{\sqrt{- \frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} + 1} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \sqrt{\operatorname{acos}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}} 
                     ___                         
                   \/ 2 *sin(x)                  
-------------------------------------------------
      ___________________     ___________________
     /        /pi      \     /     /   /    pi\\ 
2*  /  1 - sin|-- + 2*x| *  /  acos|sin|x + --|| 
  \/          \2       /  \/       \   \    2 //
2sin⁡(x)21−sin⁡(2x+π2)acos⁡(sin⁡(x+π2))\frac{\sqrt{2} \sin{\left(x \right)}}{2 \sqrt{1 - \sin{\left(2 x + \frac{\pi}{2} \right)}} \sqrt{\operatorname{acos}{\left(\sin{\left(x + \frac{\pi}{2} \right)} \right)}}} 
                                     /x\                                  
                                  cot|-|                                  
                                     \2/                                  
--------------------------------------------------------------------------
                        _____________________                             
                       /                   2          ____________________
                      /      /        2/x\\          /     /        2/x\\ 
                     /       |-1 + cot |-||         /      |-1 + cot |-|| 
/       2/x\\       /        \         \2//        /       |         \2/| 
|1 + cot |-||*     /     1 - --------------- *    /    acos|------------| 
\        \2//     /                        2     /         |       2/x\ | 
                 /            /       2/x\\     /          |1 + cot |-| | 
                /             |1 + cot |-||   \/           \        \2/ / 
              \/              \        \2//
cot⁡(x2)−(cot⁡2(x2)−1)2(cot⁡2(x2)+1)2+1(cot⁡2(x2)+1)acos⁡(cot⁡2(x2)−1cot⁡2(x2)+1)\frac{\cot{\left(\frac{x}{2} \right)}}{\sqrt{- \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 1} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \sqrt{\operatorname{acos}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}} 
                     sin(x)                     
------------------------------------------------
      __________________     ___________________
     /        2/    pi\     /     /   /    pi\\ 
2*  /  1 - sin |x + --| *  /  acos|sin|x + --|| 
  \/           \    2 /  \/       \   \    2 //
sin⁡(x)21−sin⁡2(x+π2)acos⁡(sin⁡(x+π2))\frac{\sin{\left(x \right)}}{2 \sqrt{1 - \sin^{2}{\left(x + \frac{\pi}{2} \right)}} \sqrt{\operatorname{acos}{\left(\sin{\left(x + \frac{\pi}{2} \right)} \right)}}} 
                /    pi\             
             cos|x - --|             
                \    2 /             
-------------------------------------
      ______________                 
     /    2/    pi\    ______________
2*  /  cos |x - --| *\/ acos(cos(x)) 
  \/       \    2 /
cos⁡(x−π2)2cos⁡2(x−π2)acos⁡(cos⁡(x))\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{2 \sqrt{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} \sqrt{\operatorname{acos}{\left(\cos{\left(x \right)} \right)}}} 
               /    pi\            
            cos|x - --|            
               \    2 /            
-----------------------------------
     _____________                 
    /        2       ______________
2*\/  1 - cos (x) *\/ acos(cos(x))
cos⁡(x−π2)21−cos⁡2(x)acos⁡(cos⁡(x))\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{2 \sqrt{1 - \cos^{2}{\left(x \right)}} \sqrt{\operatorname{acos}{\left(\cos{\left(x \right)} \right)}}} 
                         1                         
---------------------------------------------------
       _____________     ______________            
      /        1        /     /  1   \     /    pi\
2*   /  1 - ------- *  /  acos|------| *sec|x - --|
    /          2     \/       \sec(x)/     \    2 /
  \/        sec (x)
121−1sec⁡2(x)acos⁡(1sec⁡(x))sec⁡(x−π2)\frac{1}{2 \sqrt{1 - \frac{1}{\sec^{2}{\left(x \right)}}} \sqrt{\operatorname{acos}{\left(\frac{1}{\sec{\left(x \right)}} \right)}} \sec{\left(x - \frac{\pi}{2} \right)}} 
             sin(x)            
-------------------------------
     _________                 
    /    2       ______________
2*\/  sin (x) *\/ acos(cos(x))
sin⁡(x)2sin⁡2(x)acos⁡(cos⁡(x))\frac{\sin{\left(x \right)}}{2 \sqrt{\sin^{2}{\left(x \right)}} \sqrt{\operatorname{acos}{\left(\cos{\left(x \right)} \right)}}} 
                             1                             
-----------------------------------------------------------
        __________________       ___________________       
       /          1             /     /     1     \        
2*    /  1 - ------------ *    /  acos|-----------| *csc(x)
     /          2/pi    \     /       |   /pi    \|        
    /        csc |-- - x|    /        |csc|-- - x||        
  \/             \2     /  \/         \   \2     //
121−1csc⁡2(−x+π2)acos⁡(1csc⁡(−x+π2))csc⁡(x)\frac{1}{2 \sqrt{1 - \frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}} \sqrt{\operatorname{acos}{\left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}} \csc{\left(x \right)}} 
                                    /x\                                 
                                 tan|-|                                 
                                    \2/                                 
------------------------------------------------------------------------
                        ____________________                            
                       /                  2          ___________________
                      /      /       2/x\\          /     /       2/x\\ 
                     /       |1 - tan |-||         /      |1 - tan |-|| 
/       2/x\\       /        \        \2//        /       |        \2/| 
|1 + tan |-||*     /     1 - -------------- *    /    acos|-----------| 
\        \2//     /                       2     /         |       2/x\| 
                 /           /       2/x\\     /          |1 + tan |-|| 
                /            |1 + tan |-||   \/           \        \2// 
              \/             \        \2//
tan⁡(x2)−(1−tan⁡2(x2))2(tan⁡2(x2)+1)2+1(tan⁡2(x2)+1)acos⁡(1−tan⁡2(x2)tan⁡2(x2)+1)\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{- \frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 1} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \sqrt{\operatorname{acos}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}} 
                        1                        
-------------------------------------------------
       _________       ___________________       
      /    1          /     /     1     \        
2*   /  ------- *    /  acos|-----------| *csc(x)
    /      2        /       |   /pi    \|        
  \/    csc (x)    /        |csc|-- - x||        
                 \/         \   \2     //
121csc⁡2(x)acos⁡(1csc⁡(−x+π2))csc⁡(x)\frac{1}{2 \sqrt{\frac{1}{\csc^{2}{\left(x \right)}}} \sqrt{\operatorname{acos}{\left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}} \csc{\left(x \right)}} 
              ___                  
            \/ 2 *sin(x)           
-----------------------------------
    ______________   ______________
2*\/ 1 - cos(2*x) *\/ acos(cos(x))
2sin⁡(x)21−cos⁡(2x)acos⁡(cos⁡(x))\frac{\sqrt{2} \sin{\left(x \right)}}{2 \sqrt{1 - \cos{\left(2 x \right)}} \sqrt{\operatorname{acos}{\left(\cos{\left(x \right)} \right)}}} 
                          1                          
-----------------------------------------------------
        ______________     ______________            
       /      1           /     /  1   \     /    pi\
2*    /  ------------ *  /  acos|------| *sec|x - --|
     /      2/    pi\  \/       \sec(x)/     \    2 /
    /    sec |x - --|                                
  \/         \    2 /
121sec⁡2(x−π2)acos⁡(1sec⁡(x))sec⁡(x−π2)\frac{1}{2 \sqrt{\frac{1}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}} \sqrt{\operatorname{acos}{\left(\frac{1}{\sec{\left(x \right)}} \right)}} \sec{\left(x - \frac{\pi}{2} \right)}} 
           ___    /    pi\         
         \/ 2 *cos|x - --|         
                  \    2 /         
-----------------------------------
    ______________   ______________
2*\/ 1 - cos(2*x) *\/ acos(cos(x))
2cos⁡(x−π2)21−cos⁡(2x)acos⁡(cos⁡(x))\frac{\sqrt{2} \cos{\left(x - \frac{\pi}{2} \right)}}{2 \sqrt{1 - \cos{\left(2 x \right)}} \sqrt{\operatorname{acos}{\left(\cos{\left(x \right)} \right)}}} 
                             ___                            
                           \/ 2                             
------------------------------------------------------------
        ___________________       ___________________       
       /           1             /     /     1     \        
2*    /  1 - ------------- *    /  acos|-----------| *csc(x)
     /          /pi      \     /       |   /pi    \|        
    /        csc|-- - 2*x|    /        |csc|-- - x||        
  \/            \2       /  \/         \   \2     //
221−1csc⁡(−2x+π2)acos⁡(1csc⁡(−x+π2))csc⁡(x)\frac{\sqrt{2}}{2 \sqrt{1 - \frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}} \sqrt{\operatorname{acos}{\left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}} \csc{\left(x \right)}} 
sqrt(2)/(2*sqrt(1 - 1/csc(pi/2 - 2*x))*sqrt(acos(1/csc(pi/2 - x)))*csc(x))
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