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Simplificación de expresiones/ Descomposición en fracciones simples/ sin(x)/((tan((pi/4)-(x/2))*(1+sin(x))))

¿Cómo vas a descomponer esta sin(x)/((tan((pi/4)-(x/2))*(1+sin(x)))) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
         sin(x)         
------------------------
   /pi   x\             
tan|-- - -|*(1 + sin(x))
   \4    2/
$$\frac{\sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) \tan{\left(- \frac{x}{2} + \frac{\pi}{4} \right)}}$$ 
sin(x)/((tan(pi/4 - x/2)*(1 + sin(x))))
Simplificación general [src] 
          /x   pi\
sin(x)*tan|- + --|
          \2   4 /
------------------
    1 + sin(x)
$$\frac{\sin{\left(x \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\sin{\left(x \right)} + 1}$$ 
sin(x)*tan(x/2 + pi/4)/(1 + sin(x))
Denominador racional [src] 
        -sin(x)         
------------------------
                /x   pi\
(1 + sin(x))*tan|- - --|
                \2   4 /
$$- \frac{\sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) \tan{\left(\frac{x}{2} - \frac{\pi}{4} \right)}}$$ 
-sin(x)/((1 + sin(x))*tan(x/2 - pi/4))
Respuesta numérica [src] 
sin(x)/((1.0 + sin(x))*tan(pi/4 - x/2))
sin(x)/((1.0 + sin(x))*tan(pi/4 - x/2))
Combinatoria [src] 
         sin(x)         
------------------------
                /x   pi\
(1 + sin(x))*cot|- + --|
                \2   4 /
$$\frac{\sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) \cot{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}$$ 
sin(x)/((1 + sin(x))*cot(x/2 + pi/4))
Unión de expresiones racionales [src] 
          sin(x)          
--------------------------
                /pi - 2*x\
(1 + sin(x))*tan|--------|
                \   4    /
$$\frac{\sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) \tan{\left(\frac{\pi - 2 x}{4} \right)}}$$ 
sin(x)/((1 + sin(x))*tan((pi - 2*x)/4))
Denominador común [src] 
             sin(x)             
--------------------------------
   /x   pi\             /x   pi\
cot|- + --|*sin(x) + cot|- + --|
   \2   4 /             \2   4 /
$$\frac{\sin{\left(x \right)}}{\sin{\left(x \right)} \cot{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + \cot{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}$$ 
sin(x)/(cot(x/2 + pi/4)*sin(x) + cot(x/2 + pi/4))
Potencias [src] 
                       /   /x   pi\      /  x   pi\\      
                       | I*|- - --|    I*|- - + --||      
      /   -I*x    I*x\ |   \2   4 /      \  2   4 /|      
     -\- e     + e   /*\e           + e            /      
----------------------------------------------------------
                           /     /  x   pi\      /x   pi\\
  /      /   -I*x    I*x\\ |   I*|- - + --|    I*|- - --||
  |    I*\- e     + e   /| |     \  2   4 /      \2   4 /|
2*|1 - ------------------|*\- e             + e          /
  \            2         /
$$- \frac{\left(e^{i x} - e^{- i x}\right) \left(e^{i \left(- \frac{x}{2} + \frac{\pi}{4}\right)} + e^{i \left(\frac{x}{2} - \frac{\pi}{4}\right)}\right)}{2 \left(- \frac{i \left(e^{i x} - e^{- i x}\right)}{2} + 1\right) \left(- e^{i \left(- \frac{x}{2} + \frac{\pi}{4}\right)} + e^{i \left(\frac{x}{2} - \frac{\pi}{4}\right)}\right)}$$ 
         sin(x)         
------------------------
                /x   pi\
(1 + sin(x))*cot|- + --|
                \2   4 /
$$\frac{\sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) \cot{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}$$ 
sin(x)/((1 + sin(x))*cot(x/2 + pi/4))
Abrimos la expresión [src] 
                        sin(x)                        
------------------------------------------------------
                    /x\                     /x\       
                 cot|-|                  cot|-|*sin(x)
      1             \2/       sin(x)        \2/       
- ---------- + ---------- - ---------- + -------------
         /x\          /x\          /x\            /x\ 
  1 + cot|-|   1 + cot|-|   1 + cot|-|     1 + cot|-| 
         \2/          \2/          \2/            \2/
$$\frac{\sin{\left(x \right)}}{\frac{\sin{\left(x \right)} \cot{\left(\frac{x}{2} \right)}}{\cot{\left(\frac{x}{2} \right)} + 1} - \frac{\sin{\left(x \right)}}{\cot{\left(\frac{x}{2} \right)} + 1} + \frac{\cot{\left(\frac{x}{2} \right)}}{\cot{\left(\frac{x}{2} \right)} + 1} - \frac{1}{\cot{\left(\frac{x}{2} \right)} + 1}}$$ 
sin(x)/(-1/(1 + cot(x/2)) + cot(x/2)/(1 + cot(x/2)) - sin(x)/(1 + cot(x/2)) + cot(x/2)*sin(x)/(1 + cot(x/2)))
Parte trigonométrica [src] 
           /x\    /x   pi\     
      2*cot|-|*tan|- + --|     
           \2/    \2   4 /     
-------------------------------
              /           /x\ \
              |      2*cot|-| |
/       2/x\\ |           \2/ |
|1 + cot |-||*|1 + -----------|
\        \2// |           2/x\|
              |    1 + cot |-||
              \            \2//
$$\frac{2 \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$ 
     2/x\    4/x\    /x\    /x   pi\
8*cot |-|*sin |-|*cot|-|*tan|- + --|
      \4/     \4/    \2/    \2   4 /
------------------------------------
             1 + sin(x)
$$\frac{8 \sin^{4}{\left(\frac{x}{4} \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)} \cot^{2}{\left(\frac{x}{4} \right)} \cot{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)} + 1}$$ 
                   2/x\    /x\              
              8*cot |-|*cot|-|              
                    \4/    \2/              
--------------------------------------------
               /           /x\ \            
             2 |      2*cot|-| |            
/       2/x\\  |           \2/ |    /x   pi\
|1 + cot |-|| *|1 + -----------|*cot|- + --|
\        \4//  |           2/x\|    \2   4 /
               |    1 + cot |-||            
               \            \2//
$$\frac{8 \cot^{2}{\left(\frac{x}{4} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \cot{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}$$ 
   /pi    \
csc|-- - x|
   \2     /
-----------
   csc(x)
$$\frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}$$ 
              2/x\    /x   pi\         
         8*tan |-|*tan|- + --|         
               \4/    \2   4 /         
---------------------------------------
               /           /x\ \       
             2 |      2*tan|-| |       
/       2/x\\  |           \2/ |    /x\
|1 + tan |-|| *|1 + -----------|*tan|-|
\        \4//  |           2/x\|    \2/
               |    1 + tan |-||       
               \            \2//
$$\frac{8 \tan^{2}{\left(\frac{x}{4} \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \tan{\left(\frac{x}{2} \right)}}$$ 
               2                   
  /       2/x\\     /x\    /x   pi\
2*|1 - tan |-|| *tan|-|*tan|- + --|
  \        \4//     \2/    \2   4 /
-----------------------------------
                 /           /x\ \ 
               2 |      2*tan|-| | 
  /       2/x\\  |           \2/ | 
  |1 + tan |-|| *|1 + -----------| 
  \        \4//  |           2/x\| 
                 |    1 + tan |-|| 
                 \            \2//
$$\frac{2 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{2} \tan{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}$$ 
     4/x\    2/x\    /x   pi\
8*cos |-|*tan |-|*tan|- + --|
      \4/     \4/    \2   4 /
-----------------------------
                     /x\     
     (1 + sin(x))*tan|-|     
                     \2/
$$\frac{8 \cos^{4}{\left(\frac{x}{4} \right)} \tan^{2}{\left(\frac{x}{4} \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\sin{\left(x \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}}$$ 
     2/x\    2/pi   x\    2/x   pi\
8*sin |-|*sin |-- + -|*sin |- + --|
      \2/     \2    2/     \2   4 /
-----------------------------------
     (1 + sin(x))*cos(x)*sin(x)
$$\frac{8 \sin^{2}{\left(\frac{x}{2} \right)} \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)}}$$ 
               2                           
  /       2/x\\     4/x\    /x\    /x   pi\
2*|1 - tan |-|| *cos |-|*tan|-|*tan|- + --|
  \        \4//      \4/    \2/    \2   4 /
-------------------------------------------
                 1 + sin(x)
$$\frac{2 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{2} \cos^{4}{\left(\frac{x}{4} \right)} \tan{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\sin{\left(x \right)} + 1}$$ 
       cos(x)*sin(x)       
---------------------------
                  2/x   pi\
2*(1 + sin(x))*cos |- + --|
                   \2   4 /
$$\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{2 \left(\sin{\left(x \right)} + 1\right) \cos^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}$$ 
                /    pi\     
      sin(x)*sin|x + --|     
                \    2 /     
-----------------------------
                  2/x   3*pi\
2*(1 + sin(x))*sin |- + ----|
                   \2    4  /
$$\frac{\sin{\left(x \right)} \sin{\left(x + \frac{\pi}{2} \right)}}{2 \left(\sin{\left(x \right)} + 1\right) \sin^{2}{\left(\frac{x}{2} + \frac{3 \pi}{4} \right)}}$$ 
   /    pi\
cos|x - --|
   \    2 /
-----------
   cos(x)
$$\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}$$ 
      /    pi\    /x   pi\   
   cos|x - --|*cos|- - --|   
      \    2 /    \2   4 /   
-----------------------------
/       /    pi\\    /x   pi\
|1 + cos|x - --||*cos|- + --|
\       \    2 //    \2   4 /
$$\frac{\cos{\left(\frac{x}{2} - \frac{\pi}{4} \right)} \cos{\left(x - \frac{\pi}{2} \right)}}{\left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right) \cos{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}$$ 
     2/x\    /x\    /x   pi\
2*sin |-|*cot|-|*tan|- + --|
      \2/    \2/    \2   4 /
----------------------------
         1 + sin(x)
$$\frac{2 \sin^{2}{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)} \cot{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)} + 1}$$ 
tan(x)
$$\tan{\left(x \right)}$$ 
             /x   pi\          
          sec|- + --|          
             \2   4 /          
-------------------------------
/      1   \           /x   pi\
|1 + ------|*csc(x)*csc|- + --|
\    csc(x)/           \2   4 /
$$\frac{\sec{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)} \csc{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}$$ 
                  /x   pi\               
               sec|- + --|               
                  \2   4 /               
-----------------------------------------
/         1     \    /    pi\    /x   pi\
|1 + -----------|*sec|x - --|*sec|- - --|
|       /    pi\|    \    2 /    \2   4 /
|    sec|x - --||                        
\       \    2 //
$$\frac{\sec{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \sec{\left(\frac{x}{2} - \frac{\pi}{4} \right)} \sec{\left(x - \frac{\pi}{2} \right)}}$$ 
     2/x   pi\       
2*sin |- + --|*sin(x)
      \2   4 /       
---------------------
 (1 + sin(x))*cos(x)
$$\frac{2 \sin{\left(x \right)} \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\sin{\left(x \right)} + 1\right) \cos{\left(x \right)}}$$ 
       /    pi\    /x   pi\    
    cos|x - --|*sin|- + --|    
       \    2 /    \2   4 /    
-------------------------------
/       /    pi\\    /  x   pi\
|1 + cos|x - --||*cos|- - - --|
\       \    2 //    \  2   4 /
$$\frac{\sin{\left(\frac{x}{2} + \frac{\pi}{4} \right)} \cos{\left(x - \frac{\pi}{2} \right)}}{\left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right) \cos{\left(- \frac{x}{2} - \frac{\pi}{4} \right)}}$$ 
                       /x\                 
                  2*tan|-|                 
                       \2/                 
-------------------------------------------
              /           /x\ \            
              |      2*tan|-| |            
/       2/x\\ |           \2/ |    /x   pi\
|1 + tan |-||*|1 + -----------|*cot|- + --|
\        \2// |           2/x\|    \2   4 /
              |    1 + tan |-||            
              \            \2//
$$\frac{2 \tan{\left(\frac{x}{2} \right)}}{\left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \cot{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}$$ 
                       /x\                 
                  2*cot|-|                 
                       \2/                 
-------------------------------------------
              /           /x\ \            
              |      2*cot|-| |            
/       2/x\\ |           \2/ |    /x   pi\
|1 + cot |-||*|1 + -----------|*cot|- + --|
\        \2// |           2/x\|    \2   4 /
              |    1 + cot |-||            
              \            \2//
$$\frac{2 \cot{\left(\frac{x}{2} \right)}}{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \cot{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}$$ 
            /  x   pi\         
         csc|- - + --|         
            \  2   4 /         
-------------------------------
/      1   \           /x   pi\
|1 + ------|*csc(x)*csc|- + --|
\    csc(x)/           \2   4 /
$$\frac{\csc{\left(- \frac{x}{2} + \frac{\pi}{4} \right)}}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)} \csc{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}$$ 
  1   
------
cot(x)
$$\frac{1}{\cot{\left(x \right)}}$$ 
      /x   pi\          
   cos|- - --|*sin(x)   
      \2   4 /          
------------------------
                /x   pi\
(1 + sin(x))*cos|- + --|
                \2   4 /
$$\frac{\sin{\left(x \right)} \cos{\left(\frac{x}{2} - \frac{\pi}{4} \right)}}{\left(\sin{\left(x \right)} + 1\right) \cos{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}$$ 
            /x   pi\
2*sin(x)*tan|- + --|
            \2   4 /
--------------------
    2 + 2*sin(x)
$$\frac{2 \sin{\left(x \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{2 \sin{\left(x \right)} + 2}$$ 
          /x   pi\
sin(x)*tan|- + --|
          \2   4 /
------------------
    1 + sin(x)
$$\frac{\sin{\left(x \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\sin{\left(x \right)} + 1}$$ 
                      /x   pi\                  
                 2*sec|- + --|                  
                      \2   4 /                  
------------------------------------------------
/         1     \    /x\    /x   pi\    /x   pi\
|1 + -----------|*sec|-|*sec|- - --|*sec|- - --|
|       /    pi\|    \2/    \2   2 /    \2   4 /
|    sec|x - --||                               
\       \    2 //
$$\frac{2 \sec{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \sec{\left(\frac{x}{2} \right)} \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)} \sec{\left(\frac{x}{2} - \frac{\pi}{4} \right)}}$$ 
                2                           
  /        2/x\\     4/x\    /x\    /x   pi\
2*|-1 + cot |-|| *sin |-|*tan|-|*tan|- + --|
  \         \4//      \4/    \2/    \2   4 /
--------------------------------------------
                 1 + sin(x)
$$\frac{2 \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{x}{4} \right)} \tan{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\sin{\left(x \right)} + 1}$$ 
           /x\    /x   pi\     
      2*tan|-|*tan|- + --|     
           \2/    \2   4 /     
-------------------------------
              /           /x\ \
              |      2*tan|-| |
/       2/x\\ |           \2/ |
|1 + tan |-||*|1 + -----------|
\        \2// |           2/x\|
              |    1 + tan |-||
              \            \2//
$$\frac{2 \tan{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$ 
     /x\    /x   pi\    /x   pi\
2*cos|-|*cos|- - --|*cos|- - --|
     \2/    \2   2 /    \2   4 /
--------------------------------
 /       /    pi\\    /x   pi\  
 |1 + cos|x - --||*cos|- + --|  
 \       \    2 //    \2   4 /
$$\frac{2 \cos{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)} \cos{\left(\frac{x}{2} - \frac{\pi}{4} \right)}}{\left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right) \cos{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}$$ 
     2/x\    /x\    /x   pi\
2*cos |-|*tan|-|*tan|- + --|
      \2/    \2/    \2   4 /
----------------------------
         1 + sin(x)
$$\frac{2 \cos^{2}{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\sin{\left(x \right)} + 1}$$ 
     2   
2*sin (x)
---------
 sin(2*x)
$$\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}}$$ 
                   /  x   pi\              
              2*csc|- - + --|              
                   \  2   4 /              
-------------------------------------------
/      1   \    /x\    /pi   x\    /x   pi\
|1 + ------|*csc|-|*csc|-- - -|*csc|- + --|
\    csc(x)/    \2/    \2    2/    \2   4 /
$$\frac{2 \csc{\left(- \frac{x}{2} + \frac{\pi}{4} \right)}}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(\frac{x}{2} \right)} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)} \csc{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}$$ 
                                 2                 
                   /        2/x\\                  
                 2*|-1 + cot |-||                  
                   \         \4//                  
---------------------------------------------------
               /           /x\ \                   
             2 |      2*cot|-| |                   
/       2/x\\  |           \2/ |    /x\    /x   pi\
|1 + cot |-|| *|1 + -----------|*cot|-|*cot|- + --|
\        \4//  |           2/x\|    \2/    \2   4 /
               |    1 + cot |-||                   
               \            \2//
$$\frac{2 \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{2}}{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \cot{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}$$ 
   sec(x)  
-----------
   /    pi\
sec|x - --|
   \    2 /
$$\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}}$$ 
sec(x)/sec(x - pi/2)
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