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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
                   2      
  cos(x)        sin (x)   
---------- + -------------
cos(x) + 1               2
             (cos(x) + 1)
sin⁡2(x)(cos⁡(x)+1)2+cos⁡(x)cos⁡(x)+1\frac{\sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} + \frac{\cos{\left(x \right)}}{\cos{\left(x \right)} + 1} 
cos(x)/(cos(x) + 1) + sin(x)^2/(cos(x) + 1)^2
Simplificación general [src] 
    1     
----------
1 + cos(x)
1cos⁡(x)+1\frac{1}{\cos{\left(x \right)} + 1} 
1/(1 + cos(x))
Denominador común [src] 
            2             
     1 - sin (x) + cos(x) 
1 - ----------------------
           2              
    1 + cos (x) + 2*cos(x)
−−sin⁡2(x)+cos⁡(x)+1cos⁡2(x)+2cos⁡(x)+1+1- \frac{- \sin^{2}{\left(x \right)} + \cos{\left(x \right)} + 1}{\cos^{2}{\left(x \right)} + 2 \cos{\left(x \right)} + 1} + 1 
1 - (1 - sin(x)^2 + cos(x))/(1 + cos(x)^2 + 2*cos(x))
Combinatoria [src] 
   2         2            
cos (x) + sin (x) + cos(x)
--------------------------
                  2       
      (1 + cos(x))
sin⁡2(x)+cos⁡2(x)+cos⁡(x)(cos⁡(x)+1)2\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} + \cos{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} 
(cos(x)^2 + sin(x)^2 + cos(x))/(1 + cos(x))^2
Potencias [src] 
                   2      
  cos(x)        sin (x)   
---------- + -------------
1 + cos(x)               2
             (1 + cos(x))
cos⁡(x)cos⁡(x)+1+sin⁡2(x)(cos⁡(x)+1)2\frac{\cos{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{\sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} 
   I*x    -I*x                          
  e      e                           2  
  ---- + -----       /   -I*x    I*x\   
   2       2         \- e     + e   /   
---------------- - ---------------------
     I*x    -I*x                       2
    e      e         /     I*x    -I*x\ 
1 + ---- + -----     |    e      e    | 
     2       2     4*|1 + ---- + -----| 
                     \     2       2  /
eix2+e−ix2eix2+1+e−ix2−(eix−e−ix)24(eix2+1+e−ix2)2\frac{\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}}{\frac{e^{i x}}{2} + 1 + \frac{e^{- i x}}{2}} - \frac{\left(e^{i x} - e^{- i x}\right)^{2}}{4 \left(\frac{e^{i x}}{2} + 1 + \frac{e^{- i x}}{2}\right)^{2}} 
(exp(i*x)/2 + exp(-i*x)/2)/(1 + exp(i*x)/2 + exp(-i*x)/2) - (-exp(-i*x) + exp(i*x))^2/(4*(1 + exp(i*x)/2 + exp(-i*x)/2)^2)
Compilar la expresión [src] 
                   2      
  cos(x)        sin (x)   
---------- + -------------
1 + cos(x)               2
             (1 + cos(x))
cos⁡(x)cos⁡(x)+1+sin⁡2(x)(cos⁡(x)+1)2\frac{\cos{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{\sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} 
cos(x)/(1 + cos(x)) + sin(x)^2/(1 + cos(x))^2
Abrimos la expresión [src] 
          2                        
       sin (x)             cos(x)  
---------------------- + ----------
       2                 cos(x) + 1
1 + cos (x) + 2*cos(x)
sin⁡2(x)cos⁡2(x)+2cos⁡(x)+1+cos⁡(x)cos⁡(x)+1\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} + 2 \cos{\left(x \right)} + 1} + \frac{\cos{\left(x \right)}}{\cos{\left(x \right)} + 1} 
sin(x)^2/(1 + cos(x)^2 + 2*cos(x)) + cos(x)/(cos(x) + 1)
Unión de expresiones racionales [src] 
   2                         
sin (x) + (1 + cos(x))*cos(x)
-----------------------------
                    2        
        (1 + cos(x))
(cos⁡(x)+1)cos⁡(x)+sin⁡2(x)(cos⁡(x)+1)2\frac{\left(\cos{\left(x \right)} + 1\right) \cos{\left(x \right)} + \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} 
(sin(x)^2 + (1 + cos(x))*cos(x))/(1 + cos(x))^2
Parte trigonométrica [src] 
         1                      1          
------------------- + ---------------------
/      1   \                      2        
|1 + ------|*sec(x)   /      1   \     2   
\    sec(x)/          |1 + ------| *csc (x)
                      \    sec(x)/
1(1+1sec⁡(x))sec⁡(x)+1(1+1sec⁡(x))2csc⁡2(x)\frac{1}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(x \right)}} + \frac{1}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right)^{2} \csc^{2}{\left(x \right)}} 
         1                        1             
------------------- + --------------------------
/      1   \                      2             
|1 + ------|*sec(x)   /      1   \     2/    pi\
\    sec(x)/          |1 + ------| *sec |x - --|
                      \    sec(x)/      \    2 /
1(1+1sec⁡(x))sec⁡(x)+1(1+1sec⁡(x))2sec⁡2(x−π2)\frac{1}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(x \right)}} + \frac{1}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right)^{2} \sec^{2}{\left(x - \frac{\pi}{2} \right)}} 
                   2      
  cos(x)        sin (x)   
---------- + -------------
1 + cos(x)               2
             (1 + cos(x))
cos⁡(x)cos⁡(x)+1+sin⁡2(x)(cos⁡(x)+1)2\frac{\cos{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{\sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} 
              1                             1             
----------------------------- + --------------------------
/         1     \    /pi    \                    2        
|1 + -----------|*csc|-- - x|   /         1     \     2   
|       /pi    \|    \2     /   |1 + -----------| *csc (x)
|    csc|-- - x||               |       /pi    \|         
\       \2     //               |    csc|-- - x||         
                                \       \2     //
1(1+1csc⁡(−x+π2))csc⁡(−x+π2)+1(1+1csc⁡(−x+π2))2csc⁡2(x)\frac{1}{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{2} \csc^{2}{\left(x \right)}} 
       1        
----------------
            2/x\
    -1 + cot |-|
             \2/
1 + ------------
           2/x\ 
    1 + cot |-| 
            \2/
1cot⁡2(x2)−1cot⁡2(x2)+1+1\frac{1}{\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1} 
                  2/x\                              2/x\             
          -1 + cot |-|                         4*cot |-|             
                   \2/                               \2/             
-------------------------------- + ----------------------------------
              /            2/x\\                                    2
              |    -1 + cot |-||                  /            2/x\\ 
/       2/x\\ |             \2/|                2 |    -1 + 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(cot⁡2(x2)−1cot⁡2(x2)+1+1)(cot⁡2(x2)+1)+4cot⁡2(x2)(cot⁡2(x2)−1cot⁡2(x2)+1+1)2(cot⁡2(x2)+1)2\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} 
     /    pi\                       
  sin|x + --|             2         
     \    2 /          sin (x)      
--------------- + ------------------
       /    pi\                    2
1 + sin|x + --|   /       /    pi\\ 
       \    2 /   |1 + sin|x + --|| 
                  \       \    2 //
sin⁡(x+π2)sin⁡(x+π2)+1+sin⁡2(x)(sin⁡(x+π2)+1)2\frac{\sin{\left(x + \frac{\pi}{2} \right)}}{\sin{\left(x + \frac{\pi}{2} \right)} + 1} + \frac{\sin^{2}{\left(x \right)}}{\left(\sin{\left(x + \frac{\pi}{2} \right)} + 1\right)^{2}} 
    1     
----------
      1   
1 + ------
    sec(x)
11+1sec⁡(x)\frac{1}{1 + \frac{1}{\sec{\left(x \right)}}} 
       1       
---------------
         1     
1 + -----------
       /pi    \
    csc|-- - x|
       \2     /
11+1csc⁡(−x+π2)\frac{1}{1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}} 
                 2/x\                              2/x\            
          1 - tan |-|                         4*tan |-|            
                  \2/                               \2/            
------------------------------- + ---------------------------------
              /           2/x\\                                   2
              |    1 - tan |-||                  /           2/x\\ 
/       2/x\\ |            \2/|                2 |    1 - 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−tan⁡2(x2)tan⁡2(x2)+1+1)(tan⁡2(x2)+1)+4tan⁡2(x2)(1−tan⁡2(x2)tan⁡2(x2)+1+1)2(tan⁡2(x2)+1)2\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} 
    1     
----------
1 + cos(x)
1cos⁡(x)+1\frac{1}{\cos{\left(x \right)} + 1} 
       1       
---------------
           2/x\
    1 - tan |-|
            \2/
1 + -----------
           2/x\
    1 + tan |-|
            \2/
11−tan⁡2(x2)tan⁡2(x2)+1+1\frac{1}{\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1} 
                 2/    pi\
              cos |x - --|
  cos(x)          \    2 /
---------- + -------------
1 + cos(x)               2
             (1 + cos(x))
cos⁡(x)cos⁡(x)+1+cos⁡2(x−π2)(cos⁡(x)+1)2\frac{\cos{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} 
       1       
---------------
       /    pi\
1 + sin|x + --|
       \    2 /
1sin⁡(x+π2)+1\frac{1}{\sin{\left(x + \frac{\pi}{2} \right)} + 1} 
1/(1 + sin(x + pi/2))
Respuesta numérica [src] 
cos(x)/(1.0 + cos(x)) + sin(x)^2/(1.0 + cos(x))^2
cos(x)/(1.0 + cos(x)) + sin(x)^2/(1.0 + cos(x))^2
Denominador racional [src] 
            2             2                
(1 + cos(x)) *cos(x) + sin (x)*(1 + cos(x))
-------------------------------------------
                           3               
               (1 + cos(x))
(cos⁡(x)+1)2cos⁡(x)+(cos⁡(x)+1)sin⁡2(x)(cos⁡(x)+1)3\frac{\left(\cos{\left(x \right)} + 1\right)^{2} \cos{\left(x \right)} + \left(\cos{\left(x \right)} + 1\right) \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{3}} 
((1 + cos(x))^2*cos(x) + sin(x)^2*(1 + cos(x)))/(1 + cos(x))^3
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