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

Expresión a simplificar:

Solución

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