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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
           cos(x)          
---------------------------
    ____________   ________
2*\/ 1 - sin(x) *\/ sin(x)
cos⁡(x)21−sin⁡(x)sin⁡(x)\frac{\cos{\left(x \right)}}{2 \sqrt{1 - \sin{\left(x \right)}} \sqrt{\sin{\left(x \right)}}} 
cos(x)/(((2*sqrt(1 - sin(x)))*sqrt(sin(x))))
Denominador racional [src] 
    ____________          
 -\/ 1 - sin(x) *cos(x)   
--------------------------
                  ________
2*(-1 + sin(x))*\/ sin(x)
−1−sin⁡(x)cos⁡(x)2(sin⁡(x)−1)sin⁡(x)- \frac{\sqrt{1 - \sin{\left(x \right)}} \cos{\left(x \right)}}{2 \left(\sin{\left(x \right)} - 1\right) \sqrt{\sin{\left(x \right)}}} 
-sqrt(1 - sin(x))*cos(x)/(2*(-1 + sin(x))*sqrt(sin(x)))
Potencias [src] 
                        / I*x    -I*x\                  
                    ___ |e      e    |                  
                  \/ 2 *|---- + -----|                  
                        \ 2       2  /                  
--------------------------------------------------------
                                ________________________
     _____________________     /       /   -I*x    I*x\ 
    /    /   -I*x    I*x\     /      I*\- e     + e   / 
2*\/  -I*\- e     + e   / *  /   1 + ------------------ 
                           \/                2
2(eix2+e−ix2)2−i(eix−e−ix)i(eix−e−ix)2+1\frac{\sqrt{2} \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)}{2 \sqrt{- i \left(e^{i x} - e^{- i x}\right)} \sqrt{\frac{i \left(e^{i x} - e^{- i x}\right)}{2} + 1}} 
          cos(x)         
-------------------------
    _____________________
2*\/ (1 - sin(x))*sin(x)
cos⁡(x)2(1−sin⁡(x))sin⁡(x)\frac{\cos{\left(x \right)}}{2 \sqrt{\left(1 - \sin{\left(x \right)}\right) \sin{\left(x \right)}}} 
cos(x)/(2*sqrt((1 - sin(x))*sin(x)))
Parte trigonométrica [src] 
                        ___ /       2/x\\                      
                      \/ 2 *|1 - tan |-||                      
                            \        \2//                      
---------------------------------------------------------------
          _____________                       _________________
         /       /x\                         /            /x\  
        /     tan|-|                        /        2*tan|-|  
       /         \2/    /       2/x\\      /              \2/  
4*    /    ----------- *|1 + tan |-||*    /    1 - ----------- 
     /            2/x\  \        \2//    /                2/x\ 
    /      1 + tan |-|                  /          1 + tan |-| 
  \/               \2/                \/                   \2/
2(1−tan⁡2(x2))4tan⁡(x2)tan⁡2(x2)+11−2tan⁡(x2)tan⁡2(x2)+1(tan⁡2(x2)+1)\frac{\sqrt{2} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)}{4 \sqrt{\frac{\tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}} \sqrt{1 - \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
                  cos(x)                 
-----------------------------------------
      _________________     _____________
     /        /    pi\     /    /    pi\ 
2*  /  1 - cos|x - --| *  /  cos|x - --| 
  \/          \    2 /  \/      \    2 /
cos⁡(x)21−cos⁡(x−π2)cos⁡(x−π2)\frac{\cos{\left(x \right)}}{2 \sqrt{1 - \cos{\left(x - \frac{\pi}{2} \right)}} \sqrt{\cos{\left(x - \frac{\pi}{2} \right)}}} 
                  1                   
--------------------------------------
      ________     ____________       
     /   1        /       1           
2*  /  ------ *  /  1 - ------ *sec(x)
  \/   csc(x)  \/       csc(x)
121−1csc⁡(x)1csc⁡(x)sec⁡(x)\frac{1}{2 \sqrt{1 - \frac{1}{\csc{\left(x \right)}}} \sqrt{\frac{1}{\csc{\left(x \right)}}} \sec{\left(x \right)}} 
                     1                     
-------------------------------------------
      ________     ____________            
     /   1        /       1        /pi    \
2*  /  ------ *  /  1 - ------ *csc|-- - x|
  \/   csc(x)  \/       csc(x)     \2     /
121−1csc⁡(x)1csc⁡(x)csc⁡(−x+π2)\frac{1}{2 \sqrt{1 - \frac{1}{\csc{\left(x \right)}}} \sqrt{\frac{1}{\csc{\left(x \right)}}} \csc{\left(- x + \frac{\pi}{2} \right)}} 
                         1                          
----------------------------------------------------
        _____________       _________________       
       /      1            /          1             
2*    /  ----------- *    /  1 - ----------- *sec(x)
     /      /    pi\     /          /    pi\        
    /    sec|x - --|    /        sec|x - --|        
  \/        \    2 /  \/            \    2 /
121−1sec⁡(x−π2)1sec⁡(x−π2)sec⁡(x)\frac{1}{2 \sqrt{1 - \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}} \sqrt{\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}} \sec{\left(x \right)}} 
           /    pi\        
        sin|x + --|        
           \    2 /        
---------------------------
    ____________   ________
2*\/ 1 - sin(x) *\/ sin(x)
sin⁡(x+π2)21−sin⁡(x)sin⁡(x)\frac{\sin{\left(x + \frac{\pi}{2} \right)}}{2 \sqrt{1 - \sin{\left(x \right)}} \sqrt{\sin{\left(x \right)}}} 
                        ___ /        2/x\\                     
                      \/ 2 *|-1 + cot |-||                     
                            \         \2//                     
---------------------------------------------------------------
          _____________                       _________________
         /       /x\                         /            /x\  
        /     cot|-|                        /        2*cot|-|  
       /         \2/    /       2/x\\      /              \2/  
4*    /    ----------- *|1 + cot |-||*    /    1 - ----------- 
     /            2/x\  \        \2//    /                2/x\ 
    /      1 + cot |-|                  /          1 + cot |-| 
  \/               \2/                \/                   \2/
2(cot⁡2(x2)−1)4cot⁡(x2)cot⁡2(x2)+11−2cot⁡(x2)cot⁡2(x2)+1(cot⁡2(x2)+1)\frac{\sqrt{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{4 \sqrt{\frac{\cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}} \sqrt{1 - \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
sqrt(2)*(-1 + cot(x/2)^2)/(4*sqrt(cot(x/2)/(1 + cot(x/2)^2))*(1 + cot(x/2)^2)*sqrt(1 - 2*cot(x/2)/(1 + cot(x/2)^2)))
Respuesta numérica [src] 
0.5*(1.0 - sin(x))^(-0.5)*sin(x)^(-0.5)*cos(x)
0.5*(1.0 - sin(x))^(-0.5)*sin(x)^(-0.5)*cos(x)
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