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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
    cos(2*x)   
---------------
sin(x) + cos(x)
cos⁡(2x)sin⁡(x)+cos⁡(x)\frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}} 
cos(2*x)/(sin(x) + cos(x))
Simplificación general [src] 
  ___         
\/ 2 *cos(2*x)
--------------
     /    pi\ 
2*sin|x + --| 
     \    4 /
2cos⁡(2x)2sin⁡(x+π4)\frac{\sqrt{2} \cos{\left(2 x \right)}}{2 \sin{\left(x + \frac{\pi}{4} \right)}} 
sqrt(2)*cos(2*x)/(2*sin(x + pi/4))
Respuesta numérica [src] 
cos(2*x)/(cos(x) + sin(x))
cos(2*x)/(cos(x) + sin(x))
Potencias [src] 
          -2*I*x    2*I*x        
         e         e             
         ------- + ------        
            2        2           
---------------------------------
 I*x    -I*x     /   -I*x    I*x\
e      e       I*\- e     + e   /
---- + ----- - ------------------
 2       2             2
e2ix2+e−2ix2−i(eix−e−ix)2+eix2+e−ix2\frac{\frac{e^{2 i x}}{2} + \frac{e^{- 2 i x}}{2}}{- \frac{i \left(e^{i x} - e^{- i x}\right)}{2} + \frac{e^{i x}}{2} + \frac{e^{- i x}}{2}} 
(exp(-2*i*x)/2 + exp(2*i*x)/2)/(exp(i*x)/2 + exp(-i*x)/2 - i*(-exp(-i*x) + exp(i*x))/2)
Abrimos la expresión [src] 
                            2      
         1             2*cos (x)   
- --------------- + ---------------
  sin(x) + cos(x)   sin(x) + cos(x)
2cos⁡2(x)sin⁡(x)+cos⁡(x)−1sin⁡(x)+cos⁡(x)\frac{2 \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}} - \frac{1}{\sin{\left(x \right)} + \cos{\left(x \right)}} 
-1/(sin(x) + cos(x)) + 2*cos(x)^2/(sin(x) + cos(x))
Parte trigonométrica [src] 
  ___             /    pi\
\/ 2 *cos(2*x)*sec|x - --|
                  \    4 /
--------------------------
            2
2cos⁡(2x)sec⁡(x−π4)2\frac{\sqrt{2} \cos{\left(2 x \right)} \sec{\left(x - \frac{\pi}{4} \right)}}{2} 
               1               
-------------------------------
/  1           1     \         
|------ + -----------|*sec(2*x)
|sec(x)      /    pi\|         
|         sec|x - --||         
\            \    2 //
1(1sec⁡(x−π2)+1sec⁡(x))sec⁡(2x)\frac{1}{\left(\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(2 x \right)}} 
  ___ /       2/x   pi\\ /       2   \
\/ 2 *|1 + tan |- + --||*\1 - tan (x)/
      \        \2   8 //              
--------------------------------------
       /       2   \    /x   pi\      
     4*\1 + tan (x)/*tan|- + --|      
                        \2   8 /
2(1−tan⁡2(x))(tan⁡2(x2+π8)+1)4(tan⁡2(x)+1)tan⁡(x2+π8)\frac{\sqrt{2} \left(1 - \tan^{2}{\left(x \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right)}{4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}} 
  ___    2    /       2/x   pi\\ /       2   \
\/ 2 *cos (x)*|1 + tan |- + --||*\1 - tan (x)/
              \        \2   8 //              
----------------------------------------------
                     /x   pi\                 
                4*tan|- + --|                 
                     \2   8 /
2(1−tan⁡2(x))(tan⁡2(x2+π8)+1)cos⁡2(x)4tan⁡(x2+π8)\frac{\sqrt{2} \left(1 - \tan^{2}{\left(x \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right) \cos^{2}{\left(x \right)}}{4 \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}} 
      cos(2*x)      
--------------------
            /    pi\
cos(x) + cos|x - --|
            \    2 /
cos⁡(2x)cos⁡(x)+cos⁡(x−π2)\frac{\cos{\left(2 x \right)}}{\cos{\left(x \right)} + \cos{\left(x - \frac{\pi}{2} \right)}} 
     ___    
   \/ 2     
------------
   /     pi\
csc|-x + --|
   \     4 /
2csc⁡(−x+π4)\frac{\sqrt{2}}{\csc{\left(- x + \frac{\pi}{4} \right)}} 
  ___    /    3*pi\
\/ 2 *sin|x + ----|
         \     4  /
2sin⁡(x+3π4)\sqrt{2} \sin{\left(x + \frac{3 \pi}{4} \right)} 
                       2                  
               -1 + cot (x)               
------------------------------------------
              /        2/x\          /x\ \
              |-1 + cot |-|     2*cot|-| |
/       2   \ |         \2/          \2/ |
\1 + cot (x)/*|------------ + -----------|
              |       2/x\           2/x\|
              |1 + cot |-|    1 + cot |-||
              \        \2/            \2//
cot⁡2(x)−1(cot⁡2(x2)−1cot⁡2(x2)+1+2cot⁡(x2)cot⁡2(x2)+1)(cot⁡2(x)+1)\frac{\cot^{2}{\left(x \right)} - 1}{\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(\cot^{2}{\left(x \right)} + 1\right)} 
  ___             /    pi\
\/ 2 *cos(2*x)*csc|x + --|
                  \    4 /
--------------------------
            2
2cos⁡(2x)csc⁡(x+π4)2\frac{\sqrt{2} \cos{\left(2 x \right)} \csc{\left(x + \frac{\pi}{4} \right)}}{2} 
      /pi      \    
   sin|-- + 2*x|    
      \2       /    
--------------------
            /    pi\
sin(x) + sin|x + --|
            \    2 /
sin⁡(2x+π2)sin⁡(x)+sin⁡(x+π2)\frac{\sin{\left(2 x + \frac{\pi}{2} \right)}}{\sin{\left(x \right)} + \sin{\left(x + \frac{\pi}{2} \right)}} 
  ___    /    pi\
\/ 2 *cos|x + --|
         \    4 /
2cos⁡(x+π4)\sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)} 
  ___    /    pi\
\/ 2 *csc|x + --|
         \    4 /
-----------------
      /pi      \ 
 2*csc|-- - 2*x| 
      \2       /
2csc⁡(x+π4)2csc⁡(−2x+π2)\frac{\sqrt{2} \csc{\left(x + \frac{\pi}{4} \right)}}{2 \csc{\left(- 2 x + \frac{\pi}{2} \right)}} 
  ___    /    pi\
\/ 2 *sec|x - --|
         \    4 /
-----------------
    2*sec(2*x)
2sec⁡(x−π4)2sec⁡(2x)\frac{\sqrt{2} \sec{\left(x - \frac{\pi}{4} \right)}}{2 \sec{\left(2 x \right)}} 
  ___ /       2/x   pi\\ /        2   \
\/ 2 *|1 + cot |- + --||*\-1 + cot (x)/
      \        \2   8 //               
---------------------------------------
        /       2   \    /x   pi\      
      4*\1 + cot (x)/*cot|- + --|      
                         \2   8 /
2(cot⁡2(x)−1)(cot⁡2(x2+π8)+1)4(cot⁡2(x)+1)cot⁡(x2+π8)\frac{\sqrt{2} \left(\cot^{2}{\left(x \right)} - 1\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right)}{4 \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(\frac{x}{2} + \frac{\pi}{8} \right)}} 
  ___    2    /        2   \    /x   pi\
\/ 2 *sin (x)*\-1 + cot (x)/*tan|- + --|
                                \2   8 /
----------------------------------------
                  2/x   pi\             
             4*sin |- + --|             
                   \2   8 /
2(cot⁡2(x)−1)sin⁡2(x)tan⁡(x2+π8)4sin⁡2(x2+π8)\frac{\sqrt{2} \left(\cot^{2}{\left(x \right)} - 1\right) \sin^{2}{\left(x \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{4 \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}} 
  ___    2/x   pi\ /        2/x   pi\\
\/ 2 *sin |- + --|*|-1 + cot |- + --||
          \2   8 / \         \2   8 //
2(cot⁡2(x2+π8)−1)sin⁡2(x2+π8)\sqrt{2} \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} - 1\right) \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} 
                 1                  
------------------------------------
/  1           1     \    /pi      \
|------ + -----------|*csc|-- - 2*x|
|csc(x)      /pi    \|    \2       /
|         csc|-- - x||              
\            \2     //
1(1csc⁡(−x+π2)+1csc⁡(x))csc⁡(−2x+π2)\frac{1}{\left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(- 2 x + \frac{\pi}{2} \right)}} 
  ___ /        2/x   pi\\
\/ 2 *|-1 + cot |- + --||
      \         \2   8 //
-------------------------
            2/x   pi\    
     1 + cot |- + --|    
             \2   8 /
2(cot⁡2(x2+π8)−1)cot⁡2(x2+π8)+1\frac{\sqrt{2} \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1} 
  ___    /pi      \
\/ 2 *sin|-- + 2*x|
         \2       /
-------------------
        /    pi\   
   2*sin|x + --|   
        \    4 /
2sin⁡(2x+π2)2sin⁡(x+π4)\frac{\sqrt{2} \sin{\left(2 x + \frac{\pi}{2} \right)}}{2 \sin{\left(x + \frac{\pi}{4} \right)}} 
            1             
--------------------------
/  1        1   \         
|------ + ------|*sec(2*x)
\csc(x)   sec(x)/
1(1sec⁡(x)+1csc⁡(x))sec⁡(2x)\frac{1}{\left(\frac{1}{\sec{\left(x \right)}} + \frac{1}{\csc{\left(x \right)}}\right) \sec{\left(2 x \right)}} 
  ___         
\/ 2 *cos(2*x)
--------------
     /    pi\ 
2*sin|x + --| 
     \    4 /
2cos⁡(2x)2sin⁡(x+π4)\frac{\sqrt{2} \cos{\left(2 x \right)}}{2 \sin{\left(x + \frac{\pi}{4} \right)}} 
  ___         
\/ 2 *cos(2*x)
--------------
     /    pi\ 
2*cos|x - --| 
     \    4 /
2cos⁡(2x)2cos⁡(x−π4)\frac{\sqrt{2} \cos{\left(2 x \right)}}{2 \cos{\left(x - \frac{\pi}{4} \right)}} 
     ___   
   \/ 2    
-----------
   /    pi\
sec|x + --|
   \    4 /
2sec⁡(x+π4)\frac{\sqrt{2}}{\sec{\left(x + \frac{\pi}{4} \right)}} 
  ___ /       2/x   pi\\
\/ 2 *|1 - tan |- + --||
      \        \2   8 //
------------------------
           2/x   pi\    
    1 + tan |- + --|    
            \2   8 /
2(1−tan⁡2(x2+π8))tan⁡2(x2+π8)+1\frac{\sqrt{2} \left(1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1} 
                      2                  
               1 - tan (x)               
-----------------------------------------
              /       2/x\          /x\ \
              |1 - tan |-|     2*tan|-| |
/       2   \ |        \2/          \2/ |
\1 + tan (x)/*|----------- + -----------|
              |       2/x\          2/x\|
              |1 + tan |-|   1 + tan |-||
              \        \2/           \2//
1−tan⁡2(x)(1−tan⁡2(x2)tan⁡2(x2)+1+2tan⁡(x2)tan⁡2(x2)+1)(tan⁡2(x)+1)\frac{1 - \tan^{2}{\left(x \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(\tan^{2}{\left(x \right)} + 1\right)} 
(1 - tan(x)^2)/((1 + tan(x)^2)*((1 - tan(x/2)^2)/(1 + tan(x/2)^2) + 2*tan(x/2)/(1 + tan(x/2)^2)))
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