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Simplificación de expresiones/ Descomposición en fracciones simples/ cos(x-pi/2)^(2)*cot(x+pi/2)^(2)/(cos(x+pi/2)+cot(x-pi/2)*cot(x-pi/2)-cos(x+pi/2))

¿Cómo vas a descomponer esta cos(x-pi/2)^(2)*cot(x+pi/2)^(2)/(cos(x+pi/2)+cot(x-pi/2)*cot(x-pi/2)-cos(x+pi/2)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
                2/    pi\    2/    pi\             
             cos |x - --|*cot |x + --|             
                 \    2 /     \    2 /             
---------------------------------------------------
   /    pi\      /    pi\    /    pi\      /    pi\
cos|x + --| + cot|x - --|*cot|x - --| - cos|x + --|
   \    2 /      \    2 /    \    2 /      \    2 /
cos⁡2(x−π2)cot⁡2(x+π2)(cos⁡(x+π2)+cot⁡(x−π2)cot⁡(x−π2))−cos⁡(x+π2)\frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)} \cot^{2}{\left(x + \frac{\pi}{2} \right)}}{\left(\cos{\left(x + \frac{\pi}{2} \right)} + \cot{\left(x - \frac{\pi}{2} \right)} \cot{\left(x - \frac{\pi}{2} \right)}\right) - \cos{\left(x + \frac{\pi}{2} \right)}} 
(cos(x - pi/2)^2*cot(x + pi/2)^2)/(cos(x + pi/2) + cot(x - pi/2)*cot(x - pi/2) - cos(x + pi/2))
Descomposición de una fracción [src] 
sin(x)^2
sin⁡2(x)\sin^{2}{\left(x \right)} 
   2   
sin (x)
Simplificación general [src] 
   2   
sin (x)
sin⁡2(x)\sin^{2}{\left(x \right)} 
sin(x)^2
Respuesta numérica [src] 
cos(x - pi/2)^2*cot(x + pi/2)^2/cot(x - pi/2)^2
cos(x - pi/2)^2*cot(x + pi/2)^2/cot(x - pi/2)^2
Denominador común [src] 
   2   
sin (x)
sin⁡2(x)\sin^{2}{\left(x \right)} 
sin(x)^2
Combinatoria [src] 
   2   
sin (x)
sin⁡2(x)\sin^{2}{\left(x \right)} 
sin(x)^2
Potencias [src] 
                           2
/   /    pi\      /pi    \\ 
| I*|x - --|    I*|-- - x|| 
|   \    2 /      \2     /| 
|e             e          | 
|----------- + -----------| 
\     2             2     /
(ei(−x+π2)2+ei(x−π2)2)2\left(\frac{e^{i \left(- x + \frac{\pi}{2}\right)}}{2} + \frac{e^{i \left(x - \frac{\pi}{2}\right)}}{2}\right)^{2} 
   2   
sin (x)
sin⁡2(x)\sin^{2}{\left(x \right)} 
sin(x)^2
Abrimos la expresión [src] 
   2   
sin (x)
sin⁡2(x)\sin^{2}{\left(x \right)} 
sin(x)^2
Compilar la expresión [src] 
   2/    pi\    2/    pi\
cos |x - --|*cot |x + --|
    \    2 /     \    2 /
-------------------------
          2/    pi\      
       cot |x - --|      
           \    2 /
cos⁡2(x−π2)cot⁡2(x+π2)cot⁡2(x−π2)\frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)} \cot^{2}{\left(x + \frac{\pi}{2} \right)}}{\cot^{2}{\left(x - \frac{\pi}{2} \right)}} 
cos(x - pi/2)^2*cot(x + pi/2)^2/cot(x - pi/2)^2
Parte trigonométrica [src] 
                  2
/       2/x   pi\\ 
|1 - cot |- + --|| 
\        \2   4 // 
-------------------
                  2
/       2/x   pi\\ 
|1 + cot |- + --|| 
\        \2   4 //
(1−cot⁡2(x2+π4))2(cot⁡2(x2+π4)+1)2\frac{\left(1 - \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} 
                                4        
                   /       2/x\\     8/x\
2*(1 + cos(x)) - 4*|1 - tan |-|| *cos |-|
                   \        \4//      \4/
−4(1−tan⁡2(x4))4cos⁡8(x4)+2(cos⁡(x)+1)- 4 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{4} \cos^{8}{\left(\frac{x}{4} \right)} + 2 \left(\cos{\left(x \right)} + 1\right) 
                  2              
/         1      \              2
|1 - ------------| *(1 + sin(x)) 
|       2/x   pi\|               
|    tan |- + --||               
\        \2   4 //               
---------------------------------
                4
(1−1tan⁡2(x2+π4))2(sin⁡(x)+1)24\frac{\left(1 - \frac{1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2} \left(\sin{\left(x \right)} + 1\right)^{2}}{4} 
     1      
------------
   2/pi    \
sec |-- - x|
    \2     /
1sec⁡2(−x+π2)\frac{1}{\sec^{2}{\left(- x + \frac{\pi}{2} \right)}} 
                                     2
                    /           /x\ \ 
                  2 |      2*cot|-| | 
/       2/x   pi\\  |           \2/ | 
|1 - cot |- + --|| *|1 + -----------| 
\        \2   4 //  |           2/x\| 
                    |    1 + cot |-|| 
                    \            \2// 
--------------------------------------
                  4
(1+2cot⁡(x2)cot⁡2(x2)+1)2(1−cot⁡2(x2+π4))24\frac{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2} \left(1 - \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}\right)^{2}}{4} 
                                     2
                    /           /x\ \ 
                  2 |      2*tan|-| | 
/         1      \  |           \2/ | 
|1 - ------------| *|1 + -----------| 
|       2/x   pi\|  |           2/x\| 
|    tan |- + --||  |    1 + tan |-|| 
\        \2   4 //  \            \2// 
--------------------------------------
                  4
(1+2tan⁡(x2)tan⁡2(x2)+1)2(1−1tan⁡2(x2+π4))24\frac{\left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2} \left(1 - \frac{1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2}}{4} 
       4              4      
- ------------ + ------------
     4/pi   x\      2/pi   x\
  csc |-- - -|   csc |-- - -|
      \2    2/       \2    2/
4csc⁡2(−x2+π2)−4csc⁡4(−x2+π2)\frac{4}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - \frac{4}{\csc^{4}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} 
                  4                   2
    /        2/x\\      /        2/x\\ 
  4*|-1 + cot |-||    4*|-1 + cot |-|| 
    \         \4//      \         \4// 
- ----------------- + -----------------
                 4                   2 
    /       2/x\\       /       2/x\\  
    |1 + cot |-||       |1 + cot |-||  
    \        \4//       \        \4//
−4(cot⁡2(x4)−1)4(cot⁡2(x4)+1)4+4(cot⁡2(x4)−1)2(cot⁡2(x4)+1)2- \frac{4 \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{4}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4}} + \frac{4 \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}} 
       2/x\   
  4*cot |-|   
        \2/   
--------------
             2
/       2/x\\ 
|1 + cot |-|| 
\        \2//
4cot⁡2(x2)(cot⁡2(x2)+1)2\frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} 
     1      
------------
   2/    pi\
sec |x - --|
    \    2 /
1sec⁡2(x−π2)\frac{1}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} 
                 4                  2
    /       2/x\\      /       2/x\\ 
  4*|1 - tan |-||    4*|1 - tan |-|| 
    \        \4//      \        \4// 
- ---------------- + ----------------
                4                  2 
   /       2/x\\      /       2/x\\  
   |1 + tan |-||      |1 + tan |-||  
   \        \4//      \        \4//
−4(1−tan⁡2(x4))4(tan⁡2(x4)+1)4+4(1−tan⁡2(x4))2(tan⁡2(x4)+1)2- \frac{4 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{4}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4}} + \frac{4 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}} 
                                 4        
                   /        2/x\\     8/x\
2*(1 + cos(x)) - 4*|-1 + cot |-|| *sin |-|
                   \         \4//      \4/
2(cos⁡(x)+1)−4(cot⁡2(x4)−1)4sin⁡8(x4)2 \left(\cos{\left(x \right)} + 1\right) - 4 \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{x}{4} \right)} 
     4         4   
- ------- + -------
     4/x\      2/x\
  sec |-|   sec |-|
      \2/       \2/
4sec⁡2(x2)−4sec⁡4(x2)\frac{4}{\sec^{2}{\left(\frac{x}{2} \right)}} - \frac{4}{\sec^{4}{\left(\frac{x}{2} \right)}} 
1   cos(2*x)
- - --------
2      2
12−cos⁡(2x)2\frac{1}{2} - \frac{\cos{\left(2 x \right)}}{2} 
   2/    pi\
cos |x - --|
    \    2 /
cos⁡2(x−π2)\cos^{2}{\left(x - \frac{\pi}{2} \right)} 
       4/x\        2/x\
- 4*cos |-| + 4*cos |-|
        \2/         \2/
−4cos⁡4(x2)+4cos⁡2(x2)- 4 \cos^{4}{\left(\frac{x}{2} \right)} + 4 \cos^{2}{\left(\frac{x}{2} \right)} 
                  2
/         1      \ 
|1 - ------------| 
|       2/x   pi\| 
|    tan |- + --|| 
\        \2   4 // 
-------------------
                  2
/         1      \ 
|1 + ------------| 
|       2/x   pi\| 
|    tan |- + --|| 
\        \2   4 //
(1−1tan⁡2(x2+π4))2(1+1tan⁡2(x2+π4))2\frac{\left(1 - \frac{1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2}} 
                   2
/        2/x   pi\\ 
|-1 + tan |- + --|| 
\         \2   4 // 
--------------------
                  2 
/       2/x   pi\\  
|1 + tan |- + --||  
\        \2   4 //
(tan⁡2(x2+π4)−1)2(tan⁡2(x2+π4)+1)2\frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} 
                  2                   
/       2/x   pi\\                    
|    cos |- + --||                   2
|        \2   4 /|  /       /    pi\\ 
|1 - ------------| *|1 + cos|x - --|| 
|       2/x   pi\|  \       \    2 // 
|    cos |- - --||                    
\        \2   4 //                    
--------------------------------------
                  4
(1−cos⁡2(x2+π4)cos⁡2(x2−π4))2(cos⁡(x−π2)+1)24\frac{\left(1 - \frac{\cos^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{4} \right)}}\right)^{2} \left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right)^{2}}{4} 
                  2              
/       2/x   pi\\              2
|1 - cot |- + --|| *(1 + sin(x)) 
\        \2   4 //               
---------------------------------
                4
(1−cot⁡2(x2+π4))2(sin⁡(x)+1)24\frac{\left(1 - \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}\right)^{2} \left(\sin{\left(x \right)} + 1\right)^{2}}{4} 
                    2              
/          2       \               
|       cos (x)    |              2
|1 - --------------| *(1 + sin(x)) 
|         4/x   pi\|               
|    4*sin |- + --||               
\          \2   4 //               
-----------------------------------
                 4
(1−cos⁡2(x)4sin⁡4(x2+π4))2(sin⁡(x)+1)24\frac{\left(1 - \frac{\cos^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2} \left(\sin{\left(x \right)} + 1\right)^{2}}{4} 
   2   
sin (x)
sin⁡2(x)\sin^{2}{\left(x \right)} 
                                  2
              /        2/x   pi\ \ 
            2 |     csc |- + --| | 
/      1   \  |         \2   4 / | 
|1 + ------| *|1 - --------------| 
\    csc(x)/  |       2/  x   pi\| 
              |    csc |- - + --|| 
              \        \  2   4 // 
-----------------------------------
                 4
(1−csc⁡2(x2+π4)csc⁡2(−x2+π4))2(1+1csc⁡(x))24\frac{\left(1 - \frac{\csc^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2} \left(1 + \frac{1}{\csc{\left(x \right)}}\right)^{2}}{4} 
                                     2
                   /       2/x   pi\\ 
                 2 |    sec |- - --|| 
/         1     \  |        \2   4 /| 
|1 + -----------| *|1 - ------------| 
|       /    pi\|  |       2/x   pi\| 
|    sec|x - --||  |    sec |- + --|| 
\       \    2 //  \        \2   4 // 
--------------------------------------
                  4
(1+1sec⁡(x−π2))2(−sec⁡2(x2−π4)sec⁡2(x2+π4)+1)24\frac{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{2} \left(- \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{4} \right)}}{\sec^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}} + 1\right)^{2}}{4} 
   1   
-------
   2   
csc (x)
1csc⁡2(x)\frac{1}{\csc^{2}{\left(x \right)}} 
       4/pi   x\        2/pi   x\
- 4*sin |-- + -| + 4*sin |-- + -|
        \2    2/         \2    2/
−4sin⁡4(x2+π2)+4sin⁡2(x2+π2)- 4 \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{2} \right)} + 4 \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{2} \right)} 
       2/x\   
  4*tan |-|   
        \2/   
--------------
             2
/       2/x\\ 
|1 + tan |-|| 
\        \2//
4tan⁡2(x2)(tan⁡2(x2)+1)2\frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} 
                   2
/          1      \ 
|-1 + ------------| 
|        2/x   pi\| 
|     cot |- + --|| 
\         \2   4 // 
--------------------
                  2 
/         1      \  
|1 + ------------|  
|       2/x   pi\|  
|    cot |- + --||  
\        \2   4 //
(−1+1cot⁡2(x2+π4))2(1+1cot⁡2(x2+π4))2\frac{\left(-1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2}} 
   2            2   
cos (x - pi)*sin (x)
--------------------
         2          
      cos (x)
sin⁡2(x)cos⁡2(x−π)cos⁡2(x)\frac{\sin^{2}{\left(x \right)} \cos^{2}{\left(x - \pi \right)}}{\cos^{2}{\left(x \right)}} 
cos(x - pi)^2*sin(x)^2/cos(x)^2
Unión de expresiones racionales [src] 
   2/-pi + 2*x\    2/pi + 2*x\
cos |---------|*cot |--------|
    \    2    /     \   2    /
------------------------------
          2/-pi + 2*x\        
       cot |---------|        
           \    2    /
cos⁡2(2x−π2)cot⁡2(2x+π2)cot⁡2(2x−π2)\frac{\cos^{2}{\left(\frac{2 x - \pi}{2} \right)} \cot^{2}{\left(\frac{2 x + \pi}{2} \right)}}{\cot^{2}{\left(\frac{2 x - \pi}{2} \right)}} 
cos((-pi + 2*x)/2)^2*cot((pi + 2*x)/2)^2/cot((-pi + 2*x)/2)^2
Denominador racional [src] 
   2/    pi\    2   
cot |x + --|*sin (x)
    \    2 /        
--------------------
         2          
      tan (x)
sin⁡2(x)cot⁡2(x+π2)tan⁡2(x)\frac{\sin^{2}{\left(x \right)} \cot^{2}{\left(x + \frac{\pi}{2} \right)}}{\tan^{2}{\left(x \right)}} 
cot(x + pi/2)^2*sin(x)^2/tan(x)^2
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