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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
  cos(x)     cos(x)*sin(x)
---------- - -------------
1 + sin(x)               2
             (1 + sin(x))
−sin⁡(x)cos⁡(x)(sin⁡(x)+1)2+cos⁡(x)sin⁡(x)+1- \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}} + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)} + 1} 
cos(x)/(1 + sin(x)) - cos(x)*sin(x)/(1 + sin(x))^2
Simplificación general [src] 
    cos(x)   
-------------
            2
(1 + sin(x))
cos⁡(x)(sin⁡(x)+1)2\frac{\cos{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}} 
cos(x)/(1 + sin(x))^2
Denominador racional [src] 
            2                                    
(1 + sin(x)) *cos(x) - (1 + sin(x))*cos(x)*sin(x)
-------------------------------------------------
                              3                  
                  (1 + sin(x))
(sin⁡(x)+1)2cos⁡(x)−(sin⁡(x)+1)sin⁡(x)cos⁡(x)(sin⁡(x)+1)3\frac{\left(\sin{\left(x \right)} + 1\right)^{2} \cos{\left(x \right)} - \left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{3}} 
((1 + sin(x))^2*cos(x) - (1 + sin(x))*cos(x)*sin(x))/(1 + sin(x))^3
Respuesta numérica [src] 
cos(x)/(1.0 + sin(x)) - cos(x)*sin(x)/(1.0 + sin(x))^2
cos(x)/(1.0 + sin(x)) - cos(x)*sin(x)/(1.0 + sin(x))^2
Combinatoria [src] 
    cos(x)   
-------------
            2
(1 + sin(x))
cos⁡(x)(sin⁡(x)+1)2\frac{\cos{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}} 
cos(x)/(1 + sin(x))^2
Unión de expresiones racionales [src] 
    cos(x)   
-------------
            2
(1 + sin(x))
cos⁡(x)(sin⁡(x)+1)2\frac{\cos{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}} 
cos(x)/(1 + sin(x))^2
Denominador común [src] 
        cos(x)        
----------------------
       2              
1 + sin (x) + 2*sin(x)
cos⁡(x)sin⁡2(x)+2sin⁡(x)+1\frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)} + 2 \sin{\left(x \right)} + 1} 
cos(x)/(1 + sin(x)^2 + 2*sin(x))
Abrimos la expresión [src] 
  cos(x)         cos(x)*sin(x)     
---------- - ----------------------
1 + sin(x)          2              
             1 + sin (x) + 2*sin(x)
−sin⁡(x)cos⁡(x)sin⁡2(x)+2sin⁡(x)+1+cos⁡(x)sin⁡(x)+1- \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)} + 2 \sin{\left(x \right)} + 1} + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)} + 1} 
cos(x)/(1 + sin(x)) - cos(x)*sin(x)/(1 + sin(x)^2 + 2*sin(x))
Compilar la expresión [src] 
  cos(x)     cos(x)*sin(x)
---------- - -------------
1 + sin(x)               2
             (1 + sin(x))
cos⁡(x)sin⁡(x)+1−sin⁡(x)cos⁡(x)(sin⁡(x)+1)2\frac{\cos{\left(x \right)}}{\sin{\left(x \right)} + 1} - \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}} 
cos(x)/(1 + sin(x)) - cos(x)*sin(x)/(1 + sin(x))^2
Potencias [src] 
      I*x    -I*x          / I*x    -I*x\                 
     e      e              |e      e    | /   -I*x    I*x\
     ---- + -----        I*|---- + -----|*\- e     + e   /
      2       2            \ 2       2  /                 
---------------------- + ---------------------------------
      /   -I*x    I*x\                                2   
    I*\- e     + e   /        /      /   -I*x    I*x\\    
1 - ------------------        |    I*\- e     + e   /|    
            2               2*|1 - ------------------|    
                              \            2         /
eix2+e−ix2−i(eix−e−ix)2+1+i(eix2+e−ix2)(eix−e−ix)2(−i(eix−e−ix)2+1)2\frac{\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}}{- \frac{i \left(e^{i x} - e^{- i x}\right)}{2} + 1} + \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(e^{i x} - e^{- i x}\right)}{2} + 1\right)^{2}} 
  cos(x)     cos(x)*sin(x)
---------- - -------------
1 + sin(x)               2
             (1 + sin(x))
cos⁡(x)sin⁡(x)+1−sin⁡(x)cos⁡(x)(sin⁡(x)+1)2\frac{\cos{\left(x \right)}}{\sin{\left(x \right)} + 1} - \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}} 
cos(x)/(1 + sin(x)) - cos(x)*sin(x)/(1 + sin(x))^2
Parte trigonométrica [src] 
                            /    pi\
                  cos(x)*cos|x - --|
     cos(x)                 \    2 /
--------------- - ------------------
       /    pi\                    2
1 + cos|x - --|   /       /    pi\\ 
       \    2 /   |1 + cos|x - --|| 
                  \       \    2 //
cos⁡(x)cos⁡(x−π2)+1−cos⁡(x)cos⁡(x−π2)(cos⁡(x−π2)+1)2\frac{\cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)} + 1} - \frac{\cos{\left(x \right)} \cos{\left(x - \frac{\pi}{2} \right)}}{\left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right)^{2}} 
                  2/x\          
          -1 + cot |-|          
                   \2/          
--------------------------------
                               2
              /           /x\ \ 
              |      2*cot|-| | 
/       2/x\\ |           \2/ | 
|1 + cot |-||*|1 + -----------| 
\        \2// |           2/x\| 
              |    1 + cot |-|| 
              \            \2//
cot⁡2(x2)−1(1+2cot⁡(x2)cot⁡2(x2)+1)2(cot⁡2(x2)+1)\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
           1                              1                
------------------------ - --------------------------------
/      1   \    /pi    \               2                   
|1 + ------|*csc|-- - x|   /      1   \            /pi    \
\    csc(x)/    \2     /   |1 + ------| *csc(x)*csc|-- - x|
                           \    csc(x)/            \2     /
1(1+1csc⁡(x))csc⁡(−x+π2)−1(1+1csc⁡(x))2csc⁡(x)csc⁡(−x+π2)\frac{1}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(- x + \frac{\pi}{2} \right)}} - \frac{1}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right)^{2} \csc{\left(x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}} 
                 2/x\                     /       2/x\\    /x\     
          1 - tan |-|                   2*|1 - tan |-||*tan|-|     
                  \2/                     \        \2//    \2/     
------------------------------- - ---------------------------------
              /           /x\ \                                   2
              |      2*tan|-| |                  /           /x\ \ 
/       2/x\\ |           \2/ |                2 |      2*tan|-| | 
|1 + tan |-||*|1 + -----------|   /       2/x\\  |           \2/ | 
\        \2// |           2/x\|   |1 + tan |-|| *|1 + -----------| 
              |    1 + tan |-||   \        \2//  |           2/x\| 
              \            \2//                  |    1 + tan |-|| 
                                                 \            \2//
1−tan⁡2(x2)(1+2tan⁡(x2)tan⁡2(x2)+1)(tan⁡2(x2)+1)−2(1−tan⁡2(x2))tan⁡(x2)(1+2tan⁡(x2)tan⁡2(x2)+1)2(tan⁡2(x2)+1)2\frac{1 - \tan^{2}{\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)} - \frac{2 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \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)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} 
    /    pi\ 
 sin|x + --| 
    \    2 / 
-------------
            2
(1 + sin(x))
sin⁡(x+π2)(sin⁡(x)+1)2\frac{\sin{\left(x + \frac{\pi}{2} \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}} 
      cos(x)      
------------------
                 2
/       /    pi\\ 
|1 + cos|x - --|| 
\       \    2 //
cos⁡(x)(cos⁡(x−π2)+1)2\frac{\cos{\left(x \right)}}{\left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right)^{2}} 
           1                                 1                  
------------------------ - -------------------------------------
/         1     \                           2                   
|1 + -----------|*sec(x)   /         1     \            /    pi\
|       /    pi\|          |1 + -----------| *sec(x)*sec|x - --|
|    sec|x - --||          |       /    pi\|            \    2 /
\       \    2 //          |    sec|x - --||                    
                           \       \    2 //
1(1+1sec⁡(x−π2))sec⁡(x)−1(1+1sec⁡(x−π2))2sec⁡(x)sec⁡(x−π2)\frac{1}{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \sec{\left(x \right)}} - \frac{1}{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{2} \sec{\left(x \right)} \sec{\left(x - \frac{\pi}{2} \right)}} 
            1            
-------------------------
            2            
/      1   \     /pi    \
|1 + ------| *csc|-- - x|
\    csc(x)/     \2     /
1(1+1csc⁡(x))2csc⁡(−x+π2)\frac{1}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right)^{2} \csc{\left(- x + \frac{\pi}{2} \right)}} 
    cos(x)   
-------------
            2
(1 + sin(x))
cos⁡(x)(sin⁡(x)+1)2\frac{\cos{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}} 
         1                         1             
------------------- - ---------------------------
/      1   \                      2              
|1 + ------|*sec(x)   /      1   \               
\    csc(x)/          |1 + ------| *csc(x)*sec(x)
                      \    csc(x)/
1(1+1csc⁡(x))sec⁡(x)−1(1+1csc⁡(x))2csc⁡(x)sec⁡(x)\frac{1}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right) \sec{\left(x \right)}} - \frac{1}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right)^{2} \csc{\left(x \right)} \sec{\left(x \right)}} 
   2/x\      2/x\
cos |-| - sin |-|
    \2/       \2/
-----------------
              2  
  (1 + sin(x))
−sin⁡2(x2)+cos⁡2(x2)(sin⁡(x)+1)2\frac{- \sin^{2}{\left(\frac{x}{2} \right)} + \cos^{2}{\left(\frac{x}{2} \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}} 
                 2/x\           
          1 - tan |-|           
                  \2/           
--------------------------------
                               2
              /           /x\ \ 
              |      2*tan|-| | 
/       2/x\\ |           \2/ | 
|1 + tan |-||*|1 + -----------| 
\        \2// |           2/x\| 
              |    1 + tan |-|| 
              \            \2//
1−tan⁡2(x2)(1+2tan⁡(x2)tan⁡2(x2)+1)2(tan⁡2(x2)+1)\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
                  2/x\                   /        2/x\\    /x\     
          -1 + cot |-|                 2*|-1 + cot |-||*cot|-|     
                   \2/                   \         \2//    \2/     
------------------------------- - ---------------------------------
              /           /x\ \                                   2
              |      2*cot|-| |                  /           /x\ \ 
/       2/x\\ |           \2/ |                2 |      2*cot|-| | 
|1 + cot |-||*|1 + -----------|   /       2/x\\  |           \2/ | 
\        \2// |           2/x\|   |1 + cot |-|| *|1 + -----------| 
              |    1 + cot |-||   \        \2//  |           2/x\| 
              \            \2//                  |    1 + cot |-|| 
                                                 \            \2//
cot⁡2(x2)−1(1+2cot⁡(x2)cot⁡2(x2)+1)(cot⁡2(x2)+1)−2(cot⁡2(x2)−1)cot⁡(x2)(1+2cot⁡(x2)cot⁡2(x2)+1)2(cot⁡2(x2)+1)2\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\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)} - \frac{2 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\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)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} 
   /    pi\             /    pi\
sin|x + --|   sin(x)*sin|x + --|
   \    2 /             \    2 /
----------- - ------------------
 1 + sin(x)                 2   
                (1 + sin(x))
sin⁡(x+π2)sin⁡(x)+1−sin⁡(x)sin⁡(x+π2)(sin⁡(x)+1)2\frac{\sin{\left(x + \frac{\pi}{2} \right)}}{\sin{\left(x \right)} + 1} - \frac{\sin{\left(x \right)} \sin{\left(x + \frac{\pi}{2} \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}} 
            1            
-------------------------
                 2       
/         1     \        
|1 + -----------| *sec(x)
|       /    pi\|        
|    sec|x - --||        
\       \    2 //
1(1+1sec⁡(x−π2))2sec⁡(x)\frac{1}{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{2} \sec{\left(x \right)}} 
  cos(x)     cos(x)*sin(x)
---------- - -------------
1 + sin(x)               2
             (1 + sin(x))
cos⁡(x)sin⁡(x)+1−sin⁡(x)cos⁡(x)(sin⁡(x)+1)2\frac{\cos{\left(x \right)}}{\sin{\left(x \right)} + 1} - \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}} 
cos(x)/(1 + sin(x)) - cos(x)*sin(x)/(1 + sin(x))^2
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