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Simplificación de expresiones/ Descomposición en fracciones simples/ ctg(x)*cos(x)/(sin^2(x))

¿Cómo vas a descomponer esta ctg(x)*cos(x)/(sin^2(x)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
cot(x)*cos(x)
-------------
      2      
   sin (x)
$$\frac{\cos{\left(x \right)} \cot{\left(x \right)}}{\sin^{2}{\left(x \right)}}$$ 
(cot(x)*cos(x))/sin(x)^2
Simplificación general [src] 
   2   
cos (x)
-------
   3   
sin (x)
$$\frac{\cos^{2}{\left(x \right)}}{\sin^{3}{\left(x \right)}}$$ 
cos(x)^2/sin(x)^3
Respuesta numérica [src] 
cos(x)*cot(x)/sin(x)^2
cos(x)*cot(x)/sin(x)^2
Potencias [src] 
   / I*x    -I*x\       
   |e      e    |       
-4*|---- + -----|*cot(x)
   \ 2       2  /       
------------------------
                   2    
   /   -I*x    I*x\     
   \- e     + e   /
$$- \frac{4 \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right) \cot{\left(x \right)}}{\left(e^{i x} - e^{- i x}\right)^{2}}$$ 
-4*(exp(i*x)/2 + exp(-i*x)/2)*cot(x)/(-exp(-i*x) + exp(i*x))^2
Parte trigonométrica [src] 
               2 
  /       2/x\\  
  |1 - tan |-||  
  \        \2//  
-----------------
     2/x\    3/x\
8*cos |-|*tan |-|
      \2/     \2/
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{8 \cos^{2}{\left(\frac{x}{2} \right)} \tan^{3}{\left(\frac{x}{2} \right)}}$$ 
            1                           1            
-------------------------- - ------------------------
  /              /pi    \\     /            /pi    \\
  |           csc|-- - x||     |         csc|-- - x||
  |    1         \2     /|     |  1         \2     /|
2*|- ------ + -----------|   2*|------ + -----------|
  \  csc(x)      csc(x)  /     \csc(x)      csc(x)  /
$$- \frac{1}{2 \left(\frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} + \frac{1}{\csc{\left(x \right)}}\right)} + \frac{1}{2 \left(\frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} - \frac{1}{\csc{\left(x \right)}}\right)}$$ 
   2       3   
cos (x)*csc (x)
$$\cos^{2}{\left(x \right)} \csc^{3}{\left(x \right)}$$ 
/       2/x\\ /       2/x\\
|1 + tan |-||*|1 - tan |-||
\        \2// \        \2//
---------------------------
                  2/x\     
      4*tan(x)*tan |-|     
                   \2/
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}{4 \tan^{2}{\left(\frac{x}{2} \right)} \tan{\left(x \right)}}$$ 
cos(x)*sin(2*x)
---------------
        4      
   2*sin (x)
$$\frac{\sin{\left(2 x \right)} \cos{\left(x \right)}}{2 \sin^{4}{\left(x \right)}}$$ 
     3      
  csc (x)   
------------
   2/pi    \
csc |-- - x|
    \2     /
$$\frac{\csc^{3}{\left(x \right)}}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}$$ 
   3/    pi\
sec |x - --|
    \    2 /
------------
     2      
  sec (x)
$$\frac{\sec^{3}{\left(x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(x \right)}}$$ 
               1                                1              
------------------------------- - -----------------------------
  /                   /    pi\\     /   /    pi\              \
  |                cos|x - --||     |cos|x - --|              |
  |     /    pi\      \    2 /|     |   \    2 /      /    pi\|
2*|- cos|x - --| + -----------|   2*|----------- + cos|x - --||
  \     \    2 /      cos(x)  /     \   cos(x)        \    2 //
$$- \frac{1}{2 \left(\cos{\left(x - \frac{\pi}{2} \right)} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}\right)} + \frac{1}{2 \left(- \cos{\left(x - \frac{\pi}{2} \right)} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}\right)}$$ 
     -sin(x)     
-----------------
   2         2   
sin (x) - tan (x)
$$- \frac{\sin{\left(x \right)}}{\sin^{2}{\left(x \right)} - \tan^{2}{\left(x \right)}}$$ 
   2   
cos (x)
-------
   3   
sin (x)
$$\frac{\cos^{2}{\left(x \right)}}{\sin^{3}{\left(x \right)}}$$ 
/       2/x\\ /        2/x\\       
|1 + cot |-||*|-1 + cot |-||*cot(x)
\        \2// \         \2//       
-----------------------------------
                  2/x\             
             4*cot |-|             
                   \2/
$$\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \cot{\left(x \right)}}{4 \cot^{2}{\left(\frac{x}{2} \right)}}$$ 
               1                                1              
------------------------------- - -----------------------------
  /       1           sec(x)  \     /     1           sec(x)  \
2*|- ----------- + -----------|   2*|----------- + -----------|
  |     /    pi\      /    pi\|     |   /    pi\      /    pi\|
  |  sec|x - --|   sec|x - --||     |sec|x - --|   sec|x - --||
  \     \    2 /      \    2 //     \   \    2 /      \    2 //
$$- \frac{1}{2 \left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)} + \frac{1}{2 \left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)}$$ 
           1                        1           
----------------------- - ----------------------
  /               2   \     /     2            \
  |          2*sin (x)|     |2*sin (x)         |
2*|-sin(x) + ---------|   2*|--------- + sin(x)|
  \           sin(2*x)/     \ sin(2*x)         /
$$- \frac{1}{2 \left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + \sin{\left(x \right)}\right)} + \frac{1}{2 \left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} - \sin{\left(x \right)}\right)}$$ 
     2      
  cos (x)   
------------
   3/    pi\
cos |x - --|
    \    2 /
$$\frac{\cos^{2}{\left(x \right)}}{\cos^{3}{\left(x - \frac{\pi}{2} \right)}}$$ 
   2/    pi\
sin |x + --|
    \    2 /
------------
     3      
  sin (x)
$$\frac{\sin^{2}{\left(x + \frac{\pi}{2} \right)}}{\sin^{3}{\left(x \right)}}$$ 
           1                          1            
------------------------ - ------------------------
  /                /x\ \     /                /x\ \
  |           2*cot|-| |     |           2*cot|-| |
  |  1             \2/ |     |  1             \2/ |
2*|------ - -----------|   2*|------ + -----------|
  |cot(x)          2/x\|     |cot(x)          2/x\|
  |         1 + cot |-||     |         1 + cot |-||
  \                 \2//     \                 \2//
$$- \frac{1}{2 \left(\frac{1}{\cot{\left(x \right)}} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)} + \frac{1}{2 \left(\frac{1}{\cot{\left(x \right)}} - \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)}$$ 
            /    pi\
sin(2*x)*sin|x + --|
            \    2 /
--------------------
          4         
     2*sin (x)
$$\frac{\sin{\left(2 x \right)} \sin{\left(x + \frac{\pi}{2} \right)}}{2 \sin^{4}{\left(x \right)}}$$ 
            1                           1            
-------------------------- - ------------------------
  /         /x\          \     /       /x\          \
  |    2*tan|-|          |     |  2*tan|-|          |
  |         \2/          |     |       \2/          |
2*|- ----------- + tan(x)|   2*|----------- + tan(x)|
  |         2/x\         |     |       2/x\         |
  |  1 + tan |-|         |     |1 + tan |-|         |
  \          \2/         /     \        \2/         /
$$- \frac{1}{2 \left(\tan{\left(x \right)} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)} + \frac{1}{2 \left(\tan{\left(x \right)} - \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)}$$ 
   3   
csc (x)
-------
   2   
sec (x)
$$\frac{\csc^{3}{\left(x \right)}}{\sec^{2}{\left(x \right)}}$$ 
              2              
/        2/x\\  /       2/x\\
|-1 + cot |-|| *|1 + cot |-||
\         \2//  \        \2//
-----------------------------
               3/x\          
          8*cot |-|          
                \2/
$$\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}{8 \cot^{3}{\left(\frac{x}{2} \right)}}$$ 
         1                      1         
-------------------- - -------------------
2*(-sin(x) + tan(x))   2*(sin(x) + tan(x))
$$- \frac{1}{2 \left(\sin{\left(x \right)} + \tan{\left(x \right)}\right)} + \frac{1}{2 \left(- \sin{\left(x \right)} + \tan{\left(x \right)}\right)}$$ 
   3/pi    \
sec |-- - x|
    \2     /
------------
     2      
  sec (x)
$$\frac{\sec^{3}{\left(- x + \frac{\pi}{2} \right)}}{\sec^{2}{\left(x \right)}}$$ 
             2              
/       2/x\\  /       2/x\\
|1 - tan |-|| *|1 + tan |-||
\        \2//  \        \2//
----------------------------
              3/x\          
         8*tan |-|          
               \2/
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}{8 \tan^{3}{\left(\frac{x}{2} \right)}}$$ 
              2        
/        2/x\\     3/x\
|-1 + cot |-|| *tan |-|
\         \2//      \2/
-----------------------
            2/x\       
       8*sin |-|       
             \2/
$$\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2} \tan^{3}{\left(\frac{x}{2} \right)}}{8 \sin^{2}{\left(\frac{x}{2} \right)}}$$ 
(-1 + cot(x/2)^2)^2*tan(x/2)^3/(8*sin(x/2)^2)
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