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

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

Solución

Ha introducido [src]

/ cos(x)*sin(x) \
|---------------|
\cos(x) + sin(x)/
-----------------
1 - cos(x)*sin(x)

\frac{\sin{\left(x \right)} \cos{\left(x \right)} \frac{1}{\sin{\left(x \right)} + \cos{\left(x \right)}}}{- \sin{\left(x \right)} \cos{\left(x \right)} + 1}

((cos(x)*sin(x))/(cos(x) + sin(x)))/(1 - cos(x)*sin(x))

Simplificación general [src]

          ___                
        \/ 2 *sin(2*x)       
-----------------------------
     /    pi\      /      pi\
3*sin|x + --| + cos|3*x + --|
     \    4 /      \      4 /

\frac{\sqrt{2} \sin{\left(2 x \right)}}{3 \sin{\left(x + \frac{\pi}{4} \right)} + \cos{\left(3 x + \frac{\pi}{4} \right)}}

sqrt(2)*sin(2*x)/(3*sin(x + pi/4) + cos(3*x + pi/4))

Respuesta numérica [src]

cos(x)*sin(x)/((1.0 - cos(x)*sin(x))*(cos(x) + sin(x)))

cos(x)*sin(x)/((1.0 - cos(x)*sin(x))*(cos(x) + sin(x)))

Potencias [src]

                        / I*x    -I*x\                                       
                        |e      e    | /   -I*x    I*x\                      
                     -I*|---- + -----|*\- e     + e   /                      
                        \ 2       2  /                                       
-----------------------------------------------------------------------------
  /      / I*x    -I*x\                 \                                    
  |      |e      e    | /   -I*x    I*x\|                                    
  |    I*|---- + -----|*\- e     + e   /| / I*x    -I*x     /   -I*x    I*x\\
  |      \ 2       2  /                 | |e      e       I*\- e     + e   /|
2*|1 + ---------------------------------|*|---- + ----- - ------------------|
  \                    2                / \ 2       2             2         /

- \frac{i \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right) \left(e^{i x} - e^{- i x}\right)}{2 \left(\frac{i \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right) \left(e^{i x} - e^{- i x}\right)}{2} + 1\right) \left(- \frac{i \left(e^{i x} - e^{- i x}\right)}{2} + \frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)}

            cos(x)*sin(x)            
-------------------------------------
(1 - cos(x)*sin(x))*(cos(x) + sin(x))

\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\left(- \sin{\left(x \right)} \cos{\left(x \right)} + 1\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}

cos(x)*sin(x)/((1 - cos(x)*sin(x))*(cos(x) + sin(x)))

Denominador racional [src]

            cos(x)*sin(x)            
-------------------------------------
(1 - cos(x)*sin(x))*(cos(x) + sin(x))

\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\left(- \sin{\left(x \right)} \cos{\left(x \right)} + 1\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}

cos(x)*sin(x)/((1 - cos(x)*sin(x))*(cos(x) + sin(x)))

Combinatoria [src]

           -cos(x)*sin(x)             
--------------------------------------
(-1 + cos(x)*sin(x))*(cos(x) + sin(x))

- \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\left(\sin{\left(x \right)} \cos{\left(x \right)} - 1\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}

-cos(x)*sin(x)/((-1 + cos(x)*sin(x))*(cos(x) + sin(x)))

Unión de expresiones racionales [src]

            cos(x)*sin(x)            
-------------------------------------
(1 - cos(x)*sin(x))*(cos(x) + sin(x))

\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\left(- \sin{\left(x \right)} \cos{\left(x \right)} + 1\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}

cos(x)*sin(x)/((1 - cos(x)*sin(x))*(cos(x) + sin(x)))

Compilar la expresión [src]

            cos(x)*sin(x)            
-------------------------------------
(1 - cos(x)*sin(x))*(cos(x) + sin(x))

\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\left(- \sin{\left(x \right)} \cos{\left(x \right)} + 1\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}

cos(x)*sin(x)/((1 - cos(x)*sin(x))*(cos(x) + sin(x)))

Denominador común [src]

                 -cos(x)*sin(x)                   
--------------------------------------------------
                      2                2          
-cos(x) - sin(x) + cos (x)*sin(x) + sin (x)*cos(x)

- \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos^{2}{\left(x \right)} - \sin{\left(x \right)} - \cos{\left(x \right)}}

-cos(x)*sin(x)/(-cos(x) - sin(x) + cos(x)^2*sin(x) + sin(x)^2*cos(x))

Parte trigonométrica [src]

                           /       2/x\\    /x\                        
                         2*|1 - tan |-||*tan|-|                        
                           \        \2//    \2/                        
-----------------------------------------------------------------------
               /      /       2/x\\    /x\\ /       2/x\          /x\ \
             2 |    2*|1 - tan |-||*tan|-|| |1 - tan |-|     2*tan|-| |
/       2/x\\  |      \        \2//    \2/| |        \2/          \2/ |
|1 + tan |-|| *|1 - ----------------------|*|----------- + -----------|
\        \2//  |                     2    | |       2/x\          2/x\|
               |        /       2/x\\     | |1 + tan |-|   1 + tan |-||
               |        |1 + tan |-||     | \        \2/           \2//
               \        \        \2//     /

\frac{2 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \tan{\left(\frac{x}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(- \frac{2 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \tan{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}

                                                     ___                                                         
                                                 2*\/ 2 *sin(2*x)                                                
-----------------------------------------------------------------------------------------------------------------
                                     ___________    ___________                 ___________    ___________       
  ___                ___            /       ___    /       ___                 /       ___    /       ___        
\/ 2 *cos(3*x) + 3*\/ 2 *sin(x) - \/  2 + \/ 2  *\/  2 - \/ 2  *sin(3*x) + 3*\/  2 + \/ 2  *\/  2 - \/ 2  *cos(x)

\frac{2 \sqrt{2} \sin{\left(2 x \right)}}{3 \sqrt{2} \sin{\left(x \right)} - \sqrt{2 - \sqrt{2}} \sqrt{\sqrt{2} + 2} \sin{\left(3 x \right)} + 3 \sqrt{2 - \sqrt{2}} \sqrt{\sqrt{2} + 2} \cos{\left(x \right)} + \sqrt{2} \cos{\left(3 x \right)}}

                                1                                 
------------------------------------------------------------------
/            1         \ /  1           1     \           /    pi\
|1 - ------------------|*|------ + -----------|*sec(x)*sec|x - --|
|              /    pi\| |sec(x)      /    pi\|           \    2 /
|    sec(x)*sec|x - --|| |         sec|x - --||                   
\              \    2 // \            \    2 //

\frac{1}{\left(1 - \frac{1}{\sec{\left(x \right)} \sec{\left(x - \frac{\pi}{2} \right)}}\right) \left(\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(x \right)} \sec{\left(x - \frac{\pi}{2} \right)}}

  ___    4/x\ /       2/x\\    /x\
\/ 2 *cos |-|*|1 - tan |-||*tan|-|
          \2/ \        \2//    \2/
----------------------------------
    /    sin(2*x)\    /    pi\    
    |1 - --------|*sin|x + --|    
    \       2    /    \    4 /

\frac{\sqrt{2} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \cos^{4}{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} \right)}}{\left(1 - \frac{\sin{\left(2 x \right)}}{2}\right) \sin{\left(x + \frac{\pi}{4} \right)}}

              /         2    \       
  ___    2/x\ |      sin (x) |       
\/ 2 *sin |-|*|-1 + ---------|*sin(x)
          \2/ |          4/x\|       
              |     4*sin |-||       
              \           \2//       
-------------------------------------
       /    sin(2*x)\    /    pi\    
     2*|1 - --------|*sin|x + --|    
       \       2    /    \    4 /

\frac{\sqrt{2} \left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \sin^{2}{\left(\frac{x}{2} \right)} \sin{\left(x \right)}}{2 \left(1 - \frac{\sin{\left(2 x \right)}}{2}\right) \sin{\left(x + \frac{\pi}{4} \right)}}

  ___    3/x\ /        1   \    /x\    /    pi\
\/ 2 *sin |-|*|-1 + -------|*cos|-|*csc|x + --|
          \2/ |        2/x\|    \2/    \    4 /
              |     tan |-||                   
              \         \2//                   
-----------------------------------------------
                      sin(2*x)                 
                  1 - --------                 
                         2

\frac{\sqrt{2} \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \sin^{3}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} \csc{\left(x + \frac{\pi}{4} \right)}}{1 - \frac{\sin{\left(2 x \right)}}{2}}

         ___                
       \/ 2 *sin(2*x)       
----------------------------
  /    sin(2*x)\    /    pi\
4*|1 - --------|*sin|x + --|
  \       2    /    \    4 /

\frac{\sqrt{2} \sin{\left(2 x \right)}}{4 \left(1 - \frac{\sin{\left(2 x \right)}}{2}\right) \sin{\left(x + \frac{\pi}{4} \right)}}

                         /    pi\              
               sin(x)*sin|x + --|              
                         \    2 /              
-----------------------------------------------
/              /    pi\\ /            /    pi\\
|1 - sin(x)*sin|x + --||*|sin(x) + sin|x + --||
\              \    2 // \            \    2 //

\frac{\sin{\left(x \right)} \sin{\left(x + \frac{\pi}{2} \right)}}{\left(- \sin{\left(x \right)} \sin{\left(x + \frac{\pi}{2} \right)} + 1\right) \left(\sin{\left(x \right)} + \sin{\left(x + \frac{\pi}{2} \right)}\right)}

          ___    3/x\           /x\     
     -2*\/ 2 *sin |-|*cos(x)*cos|-|     
                  \2/           \2/     
----------------------------------------
/    sin(2*x)\                  /    pi\
|1 - --------|*(-1 + cos(x))*cos|x - --|
\       2    /                  \    4 /

- \frac{2 \sqrt{2} \sin^{3}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} \cos{\left(x \right)}}{\left(1 - \frac{\sin{\left(2 x \right)}}{2}\right) \left(\cos{\left(x \right)} - 1\right) \cos{\left(x - \frac{\pi}{4} \right)}}

                   /          2/x\   \       
                   |       cos |-|   |       
  ___    3/x   pi\ |           \2/   |    /x\
\/ 2 *cos |- - --|*|-1 + ------------|*cos|-|
          \2   2 / |        2/x   pi\|    \2/
                   |     cos |- - --||       
                   \         \2   2 //       
---------------------------------------------
       /       /      pi\\                   
       |    cos|2*x - --||                   
       |       \      2 /|    /    pi\       
       |1 - -------------|*cos|x - --|       
       \          2      /    \    4 /

\frac{\sqrt{2} \left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right) \cos{\left(\frac{x}{2} \right)} \cos^{3}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 - \frac{\cos{\left(2 x - \frac{\pi}{2} \right)}}{2}\right) \cos{\left(x - \frac{\pi}{4} \right)}}

              /       2/x   pi\\            
              |    cos |- - --||            
  ___    3/x\ |        \2   2 /|    /x   pi\
\/ 2 *cos |-|*|1 - ------------|*cos|- - --|
          \2/ |         2/x\   |    \2   2 /
              |      cos |-|   |            
              \          \2/   /            
--------------------------------------------
      /       /      pi\\                   
      |    cos|2*x - --||                   
      |       \      2 /|    /    pi\       
      |1 - -------------|*cos|x - --|       
      \          2      /    \    4 /

\frac{\sqrt{2} \left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \cos^{3}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 - \frac{\cos{\left(2 x - \frac{\pi}{2} \right)}}{2}\right) \cos{\left(x - \frac{\pi}{4} \right)}}

    ___    8/x\    4/x\ /       2/x   pi\\ /        1   \
8*\/ 2 *cos |-|*tan |-|*|1 + tan |- + --||*|-1 + -------|
            \4/     \4/ \        \2   8 // |        2/x\|
                                           |     tan |-||
                                           \         \2//
---------------------------------------------------------
            /    sin(2*x)\    /x\    /x   pi\            
            |1 - --------|*tan|-|*tan|- + --|            
            \       2    /    \2/    \2   8 /

\frac{8 \sqrt{2} \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right) \cos^{8}{\left(\frac{x}{4} \right)} \tan^{4}{\left(\frac{x}{4} \right)}}{\left(1 - \frac{\sin{\left(2 x \right)}}{2}\right) \tan{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}

       ___    /      pi\     
     \/ 2 *cos|2*x - --|     
              \      2 /     
-----------------------------
     /    pi\      /      pi\
3*cos|x - --| + cos|3*x + --|
     \    4 /      \      4 /

\frac{\sqrt{2} \cos{\left(2 x - \frac{\pi}{2} \right)}}{3 \cos{\left(x - \frac{\pi}{4} \right)} + \cos{\left(3 x + \frac{\pi}{4} \right)}}

     ___    4/x\ /       2/x   pi\\ /        1   \ 
 8*\/ 2 *tan |-|*|1 + tan |- + --||*|-1 + -------| 
             \4/ \        \2   8 // |        2/x\| 
                                    |     tan |-|| 
                                    \         \2// 
---------------------------------------------------
             4                                     
/       2/x\\  /       tan(x)  \    /x\    /x   pi\
|1 + tan |-|| *|1 - -----------|*tan|-|*tan|- + --|
\        \4//  |           2   |    \2/    \2   8 /
               \    1 + tan (x)/

\frac{8 \sqrt{2} \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right) \tan^{4}{\left(\frac{x}{4} \right)}}{\left(1 - \frac{\tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \tan{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}

  ___    4/x\ /        2/x\\    /x\
\/ 2 *sin |-|*|-1 + cot |-||*cot|-|
          \2/ \         \2//    \2/
-----------------------------------
     /    sin(2*x)\    /    pi\    
     |1 - --------|*sin|x + --|    
     \       2    /    \    4 /

\frac{\sqrt{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \sin^{4}{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(1 - \frac{\sin{\left(2 x \right)}}{2}\right) \sin{\left(x + \frac{\pi}{4} \right)}}

            cos(x)*sin(x)            
-------------------------------------
(1 - cos(x)*sin(x))*(cos(x) + sin(x))

\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\left(- \sin{\left(x \right)} \cos{\left(x \right)} + 1\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}

                        ___                          
                    2*\/ 2 *tan(x)                   
-----------------------------------------------------
              /       2/pi   3*x\         /x   pi\  \
              |1 - tan |-- + ---|    6*tan|- + --|  |
/       2   \ |        \8     2 /         \2   8 /  |
\1 + tan (x)/*|------------------ + ----------------|
              |       2/pi   3*x\          2/x   pi\|
              |1 + tan |-- + ---|   1 + tan |- + --||
              \        \8     2 /           \2   8 //

\frac{2 \sqrt{2} \tan{\left(x \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{3 x}{2} + \frac{\pi}{8} \right)}}{\tan^{2}{\left(\frac{3 x}{2} + \frac{\pi}{8} \right)} + 1} + \frac{6 \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}

                   ___                 
                 \/ 2                  
---------------------------------------
/      1               3     \         
|-------------- + -----------|*csc(2*x)
|   /       pi\      /    pi\|         
|csc|-3*x + --|   csc|x + --||         
\   \       4 /      \    4 //

\frac{\sqrt{2}}{\left(\frac{3}{\csc{\left(x + \frac{\pi}{4} \right)}} + \frac{1}{\csc{\left(- 3 x + \frac{\pi}{4} \right)}}\right) \csc{\left(2 x \right)}}

            ___    2                 
        2*\/ 2 *cos (x)*tan(x)       
-------------------------------------
       2/pi   3*x\         /x   pi\  
1 - tan |-- + ---|    6*tan|- + --|  
        \8     2 /         \2   8 /  
------------------ + ----------------
       2/pi   3*x\          2/x   pi\
1 + tan |-- + ---|   1 + tan |- + --|
        \8     2 /           \2   8 /

\frac{2 \sqrt{2} \cos^{2}{\left(x \right)} \tan{\left(x \right)}}{\frac{1 - \tan^{2}{\left(\frac{3 x}{2} + \frac{\pi}{8} \right)}}{\tan^{2}{\left(\frac{3 x}{2} + \frac{\pi}{8} \right)} + 1} + \frac{6 \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1}}

      /          2/x\   \            
      |       csc |-|   |            
  ___ |           \2/   |    /    pi\
\/ 2 *|-1 + ------------|*csc|x + --|
      |        2/pi   x\|    \    4 /
      |     csc |-- - -||            
      \         \2    2//            
-------------------------------------
 /        1     \    3/x\    /pi   x\
 |1 - ----------|*csc |-|*csc|-- - -|
 \    2*csc(2*x)/     \2/    \2    2/

\frac{\sqrt{2} \left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right) \csc{\left(x + \frac{\pi}{4} \right)}}{\left(1 - \frac{1}{2 \csc{\left(2 x \right)}}\right) \csc^{3}{\left(\frac{x}{2} \right)} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}

                   4                                                
  ___ /       2/x\\     8/x\ /       2/x   pi\\ /       2/x\\    /x\
\/ 2 *|1 - tan |-|| *cos |-|*|1 + tan |- + --||*|1 - tan |-||*tan|-|
      \        \4//      \4/ \        \2   8 // \        \2//    \2/
--------------------------------------------------------------------
                      /    sin(2*x)\    /x   pi\                    
                    2*|1 - --------|*tan|- + --|                    
                      \       2    /    \2   8 /

\frac{\sqrt{2} \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{4} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right) \cos^{8}{\left(\frac{x}{4} \right)} \tan{\left(\frac{x}{2} \right)}}{2 \left(1 - \frac{\sin{\left(2 x \right)}}{2}\right) \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}

          ___                
        \/ 2 *sin(2*x)       
-----------------------------
     /    pi\      /      pi\
3*sin|x + --| + cos|3*x + --|
     \    4 /      \      4 /

\frac{\sqrt{2} \sin{\left(2 x \right)}}{3 \sin{\left(x + \frac{\pi}{4} \right)} + \cos{\left(3 x + \frac{\pi}{4} \right)}}

    ___    3/x\           /    pi\    /x\
2*\/ 2 *cos |-|*cos(x)*csc|x + --|*sin|-|
            \2/           \    4 /    \2/
-----------------------------------------
       /    sin(2*x)\                    
       |1 - --------|*(1 + cos(x))       
       \       2    /

\frac{2 \sqrt{2} \sin{\left(\frac{x}{2} \right)} \cos^{3}{\left(\frac{x}{2} \right)} \cos{\left(x \right)} \csc{\left(x + \frac{\pi}{4} \right)}}{\left(1 - \frac{\sin{\left(2 x \right)}}{2}\right) \left(\cos{\left(x \right)} + 1\right)}

    ___    4/x\    8/x\ /        2/x\\    /x\    /x   pi\
8*\/ 2 *cot |-|*sin |-|*|-1 + cot |-||*cot|-|*tan|- + --|
            \4/     \4/ \         \2//    \2/    \2   8 /
---------------------------------------------------------
               /    sin(2*x)\    2/x   pi\               
               |1 - --------|*sin |- + --|               
               \       2    /     \2   8 /

\frac{8 \sqrt{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \sin^{8}{\left(\frac{x}{4} \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)} \cot^{4}{\left(\frac{x}{4} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(1 - \frac{\sin{\left(2 x \right)}}{2}\right) \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}

     ___    3/x\           /x\    /    pi\
-2*\/ 2 *sin |-|*cos(x)*cos|-|*sec|x - --|
             \2/           \2/    \    4 /
------------------------------------------
       /    sin(2*x)\                     
       |1 - --------|*(-1 + cos(x))       
       \       2    /

- \frac{2 \sqrt{2} \sin^{3}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} \cos{\left(x \right)} \sec{\left(x - \frac{\pi}{4} \right)}}{\left(1 - \frac{\sin{\left(2 x \right)}}{2}\right) \left(\cos{\left(x \right)} - 1\right)}

  ___    3/x\ /       2/x\\    /    pi\    /x\
\/ 2 *cos |-|*|1 - tan |-||*sec|x - --|*sin|-|
          \2/ \        \2//    \    4 /    \2/
----------------------------------------------
                     sin(2*x)                 
                 1 - --------                 
                        2

\frac{\sqrt{2} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \sin{\left(\frac{x}{2} \right)} \cos^{3}{\left(\frac{x}{2} \right)} \sec{\left(x - \frac{\pi}{4} \right)}}{1 - \frac{\sin{\left(2 x \right)}}{2}}

         ___    3/x\           /x\     
     2*\/ 2 *cos |-|*cos(x)*sin|-|     
                 \2/           \2/     
---------------------------------------
/    sin(2*x)\                 /    pi\
|1 - --------|*(1 + cos(x))*cos|x - --|
\       2    /                 \    4 /

\frac{2 \sqrt{2} \sin{\left(\frac{x}{2} \right)} \cos^{3}{\left(\frac{x}{2} \right)} \cos{\left(x \right)}}{\left(1 - \frac{\sin{\left(2 x \right)}}{2}\right) \left(\cos{\left(x \right)} + 1\right) \cos{\left(x - \frac{\pi}{4} \right)}}

                   4                                        
  ___ /       2/x\\  /       2/x   pi\\ /       2/x\\    /x\
\/ 2 *|1 - tan |-|| *|1 + tan |- + --||*|1 - tan |-||*tan|-|
      \        \4//  \        \2   8 // \        \2//    \2/
------------------------------------------------------------
                      4                                     
         /       2/x\\  /       tan(x)  \    /x   pi\       
       2*|1 + tan |-|| *|1 - -----------|*tan|- + --|       
         \        \4//  |           2   |    \2   8 /       
                        \    1 + tan (x)/

\frac{\sqrt{2} \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{4} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}}{2 \left(1 - \frac{\tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}

                         /    pi\              
               cos(x)*cos|x - --|              
                         \    2 /              
-----------------------------------------------
/              /    pi\\ /            /    pi\\
|1 - cos(x)*cos|x - --||*|cos(x) + cos|x - --||
\              \    2 // \            \    2 //

\frac{\cos{\left(x \right)} \cos{\left(x - \frac{\pi}{2} \right)}}{\left(- \cos{\left(x \right)} \cos{\left(x - \frac{\pi}{2} \right)} + 1\right) \left(\cos{\left(x \right)} + \cos{\left(x - \frac{\pi}{2} \right)}\right)}

                     ___                   
                   \/ 2                    
-------------------------------------------
/      1              3     \    /      pi\
|------------- + -----------|*sec|2*x - --|
|   /      pi\      /    pi\|    \      2 /
|sec|3*x + --|   sec|x - --||              
\   \      4 /      \    4 //

\frac{\sqrt{2}}{\left(\frac{1}{\sec{\left(3 x + \frac{\pi}{4} \right)}} + \frac{3}{\sec{\left(x - \frac{\pi}{4} \right)}}\right) \sec{\left(2 x - \frac{\pi}{2} \right)}}

         /         2/x\   \              
         |      sec |-|   |              
     ___ |          \2/   |    /    pi\  
   \/ 2 *|1 - ------------|*sec|x - --|  
         |       2/x   pi\|    \    4 /  
         |    sec |- - --||              
         \        \2   2 //              
-----------------------------------------
/           1       \    3/x\    /x   pi\
|1 - ---------------|*sec |-|*sec|- - --|
|         /      pi\|     \2/    \2   2 /
|    2*sec|2*x - --||                    
\         \      2 //

\frac{\sqrt{2} \left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(x - \frac{\pi}{4} \right)}}{\left(1 - \frac{1}{2 \sec{\left(2 x - \frac{\pi}{2} \right)}}\right) \sec^{3}{\left(\frac{x}{2} \right)} \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}

                    4                                         
  ___ /        2/x\\     8/x\ /       2/x\\    /x\    /x   pi\
\/ 2 *|-1 + cot |-|| *sin |-|*|1 - tan |-||*tan|-|*tan|- + --|
      \         \4//      \4/ \        \2//    \2/    \2   8 /
--------------------------------------------------------------
                  /    sin(2*x)\    2/x   pi\                 
                2*|1 - --------|*sin |- + --|                 
                  \       2    /     \2   8 /

\frac{\sqrt{2} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{x}{4} \right)} \tan{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{2 \left(1 - \frac{\sin{\left(2 x \right)}}{2}\right) \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}

        /        2/x   pi\\              
        |     sec |- - --||              
    ___ |         \2   2 /|    /    pi\  
  \/ 2 *|-1 + ------------|*sec|x - --|  
        |          2/x\   |    \    4 /  
        |       sec |-|   |              
        \           \2/   /              
-----------------------------------------
/           1       \    /x\    3/x   pi\
|1 - ---------------|*sec|-|*sec |- - --|
|         /      pi\|    \2/     \2   2 /
|    2*sec|2*x - --||                    
\         \      2 //

\frac{\sqrt{2} \left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right) \sec{\left(x - \frac{\pi}{4} \right)}}{\left(1 - \frac{1}{2 \sec{\left(2 x - \frac{\pi}{2} \right)}}\right) \sec{\left(\frac{x}{2} \right)} \sec^{3}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}

                             /         4/x\\
                             |    4*sin |-||
    ___    2/x\    4/pi   x\ |          \2/|
2*\/ 2 *sin |-|*sin |-- + -|*|1 - ---------|
            \2/     \2    2/ |        2    |
                             \     sin (x) /
--------------------------------------------
     /    sin(2*x)\           /    pi\      
     |1 - --------|*sin(x)*sin|x + --|      
     \       2    /           \    4 /

\frac{2 \sqrt{2} \left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \sin^{2}{\left(\frac{x}{2} \right)} \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{\left(1 - \frac{\sin{\left(2 x \right)}}{2}\right) \sin{\left(x \right)} \sin{\left(x + \frac{\pi}{4} \right)}}

          ___                
        \/ 2 *sin(2*x)       
-----------------------------
     /    pi\      /      pi\
3*cos|x - --| + cos|3*x + --|
     \    4 /      \      4 /

\frac{\sqrt{2} \sin{\left(2 x \right)}}{3 \cos{\left(x - \frac{\pi}{4} \right)} + \cos{\left(3 x + \frac{\pi}{4} \right)}}

    ___    4/x\ /       2/x   pi\\ /        2/x\\    /x\
8*\/ 2 *cot |-|*|1 + cot |- + --||*|-1 + cot |-||*cot|-|
            \4/ \        \2   8 // \         \2//    \2/
--------------------------------------------------------
                   4                                    
      /       2/x\\  /       cot(x)  \    /x   pi\      
      |1 + cot |-|| *|1 - -----------|*cot|- + --|      
      \        \4//  |           2   |    \2   8 /      
                     \    1 + cot (x)/

\frac{8 \sqrt{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right) \cot^{4}{\left(\frac{x}{4} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(1 - \frac{\cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \cot{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}

                                1                                 
------------------------------------------------------------------
/            1         \ /  1           1     \           /pi    \
|1 - ------------------|*|------ + -----------|*csc(x)*csc|-- - x|
|              /pi    \| |csc(x)      /pi    \|           \2     /
|    csc(x)*csc|-- - x|| |         csc|-- - x||                   
\              \2     // \            \2     //

\frac{1}{\left(1 - \frac{1}{\csc{\left(x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}}\right) \left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}}

           ___                 
         \/ 2 *sin(2*x)        
-------------------------------
     /    pi\      /      3*pi\
3*sin|x + --| + sin|3*x + ----|
     \    4 /      \       4  /

\frac{\sqrt{2} \sin{\left(2 x \right)}}{3 \sin{\left(x + \frac{\pi}{4} \right)} + \sin{\left(3 x + \frac{3 \pi}{4} \right)}}

                        ___                           
                    2*\/ 2 *cot(x)                    
------------------------------------------------------
              /        2/pi   3*x\         /x   pi\  \
              |-1 + cot |-- + ---|    6*cot|- + --|  |
/       2   \ |         \8     2 /         \2   8 /  |
\1 + cot (x)/*|------------------- + ----------------|
              |        2/pi   3*x\          2/x   pi\|
              | 1 + cot |-- + ---|   1 + cot |- + --||
              \         \8     2 /           \2   8 //

\frac{2 \sqrt{2} \cot{\left(x \right)}}{\left(\frac{\cot^{2}{\left(\frac{3 x}{2} + \frac{\pi}{8} \right)} - 1}{\cot^{2}{\left(\frac{3 x}{2} + \frac{\pi}{8} \right)} + 1} + \frac{6 \cot{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1}\right) \left(\cot^{2}{\left(x \right)} + 1\right)}

      /       2/pi   x\\            
      |    csc |-- - -||            
  ___ |        \2    2/|    /    pi\
\/ 2 *|1 - ------------|*csc|x + --|
      |         2/x\   |    \    4 /
      |      csc |-|   |            
      \          \2/   /            
------------------------------------
/        1     \    /x\    3/pi   x\
|1 - ----------|*csc|-|*csc |-- - -|
\    2*csc(2*x)/    \2/     \2    2/

\frac{\sqrt{2} \left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc{\left(x + \frac{\pi}{4} \right)}}{\left(1 - \frac{1}{2 \csc{\left(2 x \right)}}\right) \csc{\left(\frac{x}{2} \right)} \csc^{3}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}

                         1                         
---------------------------------------------------
/          1      \ /  1        1   \              
|1 - -------------|*|------ + ------|*csc(x)*sec(x)
\    csc(x)*sec(x)/ \csc(x)   sec(x)/

\frac{1}{\left(1 - \frac{1}{\csc{\left(x \right)} \sec{\left(x \right)}}\right) \left(\frac{1}{\sec{\left(x \right)}} + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)} \sec{\left(x \right)}}

                           /        2/x\\    /x\                         
                         2*|-1 + cot |-||*cot|-|                         
                           \         \2//    \2/                         
-------------------------------------------------------------------------
               /      /        2/x\\    /x\\ /        2/x\          /x\ \
             2 |    2*|-1 + cot |-||*cot|-|| |-1 + cot |-|     2*cot|-| |
/       2/x\\  |      \         \2//    \2/| |         \2/          \2/ |
|1 + cot |-|| *|1 - -----------------------|*|------------ + -----------|
\        \2//  |                      2    | |       2/x\           2/x\|
               |         /       2/x\\     | |1 + cot |-|    1 + cot |-||
               |         |1 + cot |-||     | \        \2/            \2//
               \         \        \2//     /

\frac{2 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \cot{\left(\frac{x}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(- \frac{2 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \cot{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}

                    4                                 
  ___ /        2/x\\  /       2/x   pi\\ /       1   \
\/ 2 *|-1 + cot |-|| *|1 + cot |- + --||*|1 - -------|
      \         \4//  \        \2   8 // |       2/x\|
                                         |    cot |-||
                                         \        \2//
------------------------------------------------------
               4                                      
  /       2/x\\  /       cot(x)  \    /x\    /x   pi\ 
2*|1 + cot |-|| *|1 - -----------|*cot|-|*cot|- + --| 
  \        \4//  |           2   |    \2/    \2   8 / 
                 \    1 + cot (x)/

\frac{\sqrt{2} \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{4} \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right)}{2 \left(1 - \frac{\cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \cot{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}

sqrt(2)*(-1 + cot(x/4)^2)^4*(1 + cot(x/2 + pi/8)^2)*(1 - 1/cot(x/2)^2)/(2*(1 + cot(x/4)^2)^4*(1 - cot(x)/(1 + cot(x)^2))*cot(x/2)*cot(x/2 + pi/8))

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

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