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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
  cos(2*x)     1 + tan(x)
------------ - ----------
1 - sin(2*x)   1 - tan(x)
−tan⁡(x)+11−tan⁡(x)+cos⁡(2x)1−sin⁡(2x)- \frac{\tan{\left(x \right)} + 1}{1 - \tan{\left(x \right)}} + \frac{\cos{\left(2 x \right)}}{1 - \sin{\left(2 x \right)}} 
cos(2*x)/(1 - sin(2*x)) - (1 + tan(x))/(1 - tan(x))
Simplificación general [src] 
(1 + tan(x))*(-1 + sin(2*x)) - (-1 + tan(x))*cos(2*x)
-----------------------------------------------------
            (-1 + sin(2*x))*(-1 + tan(x))
(sin⁡(2x)−1)(tan⁡(x)+1)−(tan⁡(x)−1)cos⁡(2x)(sin⁡(2x)−1)(tan⁡(x)−1)\frac{\left(\sin{\left(2 x \right)} - 1\right) \left(\tan{\left(x \right)} + 1\right) - \left(\tan{\left(x \right)} - 1\right) \cos{\left(2 x \right)}}{\left(\sin{\left(2 x \right)} - 1\right) \left(\tan{\left(x \right)} - 1\right)} 
((1 + tan(x))*(-1 + sin(2*x)) - (-1 + tan(x))*cos(2*x))/((-1 + sin(2*x))*(-1 + tan(x)))
Respuesta numérica [src] 
cos(2*x)/(1.0 - sin(2*x)) - (1.0 + tan(x))/(1.0 - tan(x))
cos(2*x)/(1.0 - sin(2*x)) - (1.0 + tan(x))/(1.0 - tan(x))
Potencias [src] 
                                   /   I*x    -I*x\
      -2*I*x    2*I*x            I*\- e    + e    /
     e         e             1 + ------------------
     ------- + ------                I*x    -I*x   
        2        2                  e    + e       
-------------------------- - ----------------------
      /   -2*I*x    2*I*x\         /   I*x    -I*x\
    I*\- e       + e     /       I*\- e    + e    /
1 + ----------------------   1 - ------------------
              2                      I*x    -I*x   
                                    e    + e
−i(−eix+e−ix)eix+e−ix+1−i(−eix+e−ix)eix+e−ix+1+e2ix2+e−2ix2i(e2ix−e−2ix)2+1- \frac{\frac{i \left(- e^{i x} + e^{- i x}\right)}{e^{i x} + e^{- i x}} + 1}{- \frac{i \left(- e^{i x} + e^{- i x}\right)}{e^{i x} + e^{- i x}} + 1} + \frac{\frac{e^{2 i x}}{2} + \frac{e^{- 2 i x}}{2}}{\frac{i \left(e^{2 i x} - e^{- 2 i x}\right)}{2} + 1} 
  cos(2*x)     -1 - tan(x)
------------ + -----------
1 - sin(2*x)    1 - tan(x)
−tan⁡(x)−11−tan⁡(x)+cos⁡(2x)1−sin⁡(2x)\frac{- \tan{\left(x \right)} - 1}{1 - \tan{\left(x \right)}} + \frac{\cos{\left(2 x \right)}}{1 - \sin{\left(2 x \right)}} 
cos(2*x)/(1 - sin(2*x)) + (-1 - tan(x))/(1 - tan(x))
Denominador racional [src] 
(1 - sin(2*x))*(-1 - tan(x)) + (1 - tan(x))*cos(2*x)
----------------------------------------------------
            (1 - sin(2*x))*(1 - tan(x))
(1−sin⁡(2x))(−tan⁡(x)−1)+(1−tan⁡(x))cos⁡(2x)(1−sin⁡(2x))(1−tan⁡(x))\frac{\left(1 - \sin{\left(2 x \right)}\right) \left(- \tan{\left(x \right)} - 1\right) + \left(1 - \tan{\left(x \right)}\right) \cos{\left(2 x \right)}}{\left(1 - \sin{\left(2 x \right)}\right) \left(1 - \tan{\left(x \right)}\right)} 
((1 - sin(2*x))*(-1 - tan(x)) + (1 - tan(x))*cos(2*x))/((1 - sin(2*x))*(1 - tan(x)))
Combinatoria [src] 
-(1 - cos(2*x) - sin(2*x) + cos(2*x)*tan(x) - sin(2*x)*tan(x) + tan(x)) 
------------------------------------------------------------------------
                     (-1 + sin(2*x))*(-1 + tan(x))
−−sin⁡(2x)tan⁡(x)−sin⁡(2x)+cos⁡(2x)tan⁡(x)−cos⁡(2x)+tan⁡(x)+1(sin⁡(2x)−1)(tan⁡(x)−1)- \frac{- \sin{\left(2 x \right)} \tan{\left(x \right)} - \sin{\left(2 x \right)} + \cos{\left(2 x \right)} \tan{\left(x \right)} - \cos{\left(2 x \right)} + \tan{\left(x \right)} + 1}{\left(\sin{\left(2 x \right)} - 1\right) \left(\tan{\left(x \right)} - 1\right)} 
-(1 - cos(2*x) - sin(2*x) + cos(2*x)*tan(x) - sin(2*x)*tan(x) + tan(x))/((-1 + sin(2*x))*(-1 + tan(x)))
Unión de expresiones racionales [src] 
(1 - tan(x))*cos(2*x) - (1 - sin(2*x))*(1 + tan(x))
---------------------------------------------------
            (1 - sin(2*x))*(1 - tan(x))
−(1−sin⁡(2x))(tan⁡(x)+1)+(1−tan⁡(x))cos⁡(2x)(1−sin⁡(2x))(1−tan⁡(x))\frac{- \left(1 - \sin{\left(2 x \right)}\right) \left(\tan{\left(x \right)} + 1\right) + \left(1 - \tan{\left(x \right)}\right) \cos{\left(2 x \right)}}{\left(1 - \sin{\left(2 x \right)}\right) \left(1 - \tan{\left(x \right)}\right)} 
((1 - tan(x))*cos(2*x) - (1 - sin(2*x))*(1 + tan(x)))/((1 - sin(2*x))*(1 - tan(x)))
Denominador común [src] 
    2 - cos(2*x) - 2*sin(2*x) + cos(2*x)*tan(x)
1 - -------------------------------------------
      1 - sin(2*x) - tan(x) + sin(2*x)*tan(x)
1−−2sin⁡(2x)+cos⁡(2x)tan⁡(x)−cos⁡(2x)+2sin⁡(2x)tan⁡(x)−sin⁡(2x)−tan⁡(x)+11 - \frac{- 2 \sin{\left(2 x \right)} + \cos{\left(2 x \right)} \tan{\left(x \right)} - \cos{\left(2 x \right)} + 2}{\sin{\left(2 x \right)} \tan{\left(x \right)} - \sin{\left(2 x \right)} - \tan{\left(x \right)} + 1} 
1 - (2 - cos(2*x) - 2*sin(2*x) + cos(2*x)*tan(x))/(1 - sin(2*x) - tan(x) + sin(2*x)*tan(x))
Parte trigonométrica [src] 
           2                   
      2*sin (x)      /pi      \
  1 + ---------   sin|-- + 2*x|
       sin(2*x)      \2       /
- ------------- - -------------
           2      -1 + sin(2*x)
      2*sin (x)                
  1 - ---------                
       sin(2*x)
−sin⁡(2x+π2)sin⁡(2x)−1−2sin⁡2(x)sin⁡(2x)+1−2sin⁡2(x)sin⁡(2x)+1- \frac{\sin{\left(2 x + \frac{\pi}{2} \right)}}{\sin{\left(2 x \right)} - 1} - \frac{\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 1}{- \frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 1} 
                              sec(x)
                          1 + ------
           1                  csc(x)
----------------------- - ----------
/       1    \                sec(x)
|1 - --------|*sec(2*x)   1 - ------
\    csc(2*x)/                csc(x)
1(1−1csc⁡(2x))sec⁡(2x)−1+sec⁡(x)csc⁡(x)1−sec⁡(x)csc⁡(x)\frac{1}{\left(1 - \frac{1}{\csc{\left(2 x \right)}}\right) \sec{\left(2 x \right)}} - \frac{1 + \frac{\sec{\left(x \right)}}{\csc{\left(x \right)}}}{1 - \frac{\sec{\left(x \right)}}{\csc{\left(x \right)}}} 
        1                                      
  1 + ------                     2             
      cot(x)             -1 + cot (x)          
- ---------- - --------------------------------
        1      /       2   \ /       2*cot(x) \
  1 - ------   \1 + cot (x)/*|-1 + -----------|
      cot(x)                 |            2   |
                             \     1 + cot (x)/
−1+1cot⁡(x)1−1cot⁡(x)−cot⁡2(x)−1(−1+2cot⁡(x)cot⁡2(x)+1)(cot⁡2(x)+1)- \frac{1 + \frac{1}{\cot{\left(x \right)}}}{1 - \frac{1}{\cot{\left(x \right)}}} - \frac{\cot^{2}{\left(x \right)} - 1}{\left(-1 + \frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right) \left(\cot^{2}{\left(x \right)} + 1\right)} 
                           /    pi\
                        cos|x - --|
                           \    2 /
                    1 + -----------
     cos(2*x)              cos(x)  
----------------- - ---------------
       /      pi\          /    pi\
1 - cos|2*x - --|       cos|x - --|
       \      2 /          \    2 /
                    1 - -----------
                           cos(x)
cos⁡(2x)1−cos⁡(2x−π2)−1+cos⁡(x−π2)cos⁡(x)1−cos⁡(x−π2)cos⁡(x)\frac{\cos{\left(2 x \right)}}{1 - \cos{\left(2 x - \frac{\pi}{2} \right)}} - \frac{1 + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}}{1 - \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}} 
  1 + tan(x)      cos(2*x)  
- ---------- - -------------
  1 - tan(x)   -1 + sin(2*x)
−cos⁡(2x)sin⁡(2x)−1−tan⁡(x)+11−tan⁡(x)- \frac{\cos{\left(2 x \right)}}{\sin{\left(2 x \right)} - 1} - \frac{\tan{\left(x \right)} + 1}{1 - \tan{\left(x \right)}} 
         sec(x)                                  
  1 + -----------                                
         /    pi\                                
      sec|x - --|                                
         \    2 /                 1              
- --------------- - -----------------------------
         sec(x)     /           1      \         
  1 - -----------   |-1 + -------------|*sec(2*x)
         /    pi\   |        /      pi\|         
      sec|x - --|   |     sec|2*x - --||         
         \    2 /   \        \      2 //
−sec⁡(x)sec⁡(x−π2)+1−sec⁡(x)sec⁡(x−π2)+1−1(−1+1sec⁡(2x−π2))sec⁡(2x)- \frac{\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + 1}{- \frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + 1} - \frac{1}{\left(-1 + \frac{1}{\sec{\left(2 x - \frac{\pi}{2} \right)}}\right) \sec{\left(2 x \right)}} 
         /    pi\                     
      cos|x - --|                     
         \    2 /                     
  1 + -----------                     
         cos(x)          cos(2*x)     
- --------------- - ------------------
         /    pi\           /      pi\
      cos|x - --|   -1 + cos|2*x - --|
         \    2 /           \      2 /
  1 - -----------                     
         cos(x)
−cos⁡(2x)cos⁡(2x−π2)−1−1+cos⁡(x−π2)cos⁡(x)1−cos⁡(x−π2)cos⁡(x)- \frac{\cos{\left(2 x \right)}}{\cos{\left(2 x - \frac{\pi}{2} \right)} - 1} - \frac{1 + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}}{1 - \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}} 
                   sin(x)
               1 + ------
  cos(2*x)         cos(x)
------------ - ----------
1 - sin(2*x)       sin(x)
               1 - ------
                   cos(x)
−sin⁡(x)cos⁡(x)+1−sin⁡(x)cos⁡(x)+1+cos⁡(2x)1−sin⁡(2x)- \frac{\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 1}{- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 1} + \frac{\cos{\left(2 x \right)}}{1 - \sin{\left(2 x \right)}} 
                                      /pi    \
                                   csc|-- - x|
                                      \2     /
                               1 + -----------
             1                        csc(x)  
---------------------------- - ---------------
/       1    \    /pi      \          /pi    \
|1 - --------|*csc|-- - 2*x|       csc|-- - x|
\    csc(2*x)/    \2       /          \2     /
                               1 - -----------
                                      csc(x)
1(1−1csc⁡(2x))csc⁡(−2x+π2)−1+csc⁡(−x+π2)csc⁡(x)1−csc⁡(−x+π2)csc⁡(x)\frac{1}{\left(1 - \frac{1}{\csc{\left(2 x \right)}}\right) \csc{\left(- 2 x + \frac{\pi}{2} \right)}} - \frac{1 + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}}{1 - \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}} 
                        2   
                   2*sin (x)
               1 + ---------
  cos(2*x)          sin(2*x)
------------ - -------------
1 - sin(2*x)            2   
                   2*sin (x)
               1 - ---------
                    sin(2*x)
−2sin⁡2(x)sin⁡(2x)+1−2sin⁡2(x)sin⁡(2x)+1+cos⁡(2x)1−sin⁡(2x)- \frac{\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 1}{- \frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 1} + \frac{\cos{\left(2 x \right)}}{1 - \sin{\left(2 x \right)}} 
         /pi    \                                
      csc|-- - x|                                
         \2     /                                
  1 + -----------                                
         csc(x)                   1              
- --------------- - -----------------------------
         /pi    \   /        1    \    /pi      \
      csc|-- - x|   |-1 + --------|*csc|-- - 2*x|
         \2     /   \     csc(2*x)/    \2       /
  1 - -----------                                
         csc(x)
−1+csc⁡(−x+π2)csc⁡(x)1−csc⁡(−x+π2)csc⁡(x)−1(−1+1csc⁡(2x))csc⁡(−2x+π2)- \frac{1 + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}}{1 - \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}} - \frac{1}{\left(-1 + \frac{1}{\csc{\left(2 x \right)}}\right) \csc{\left(- 2 x + \frac{\pi}{2} \right)}} 
                                      sec(x)  
                               1 + -----------
                                      /    pi\
                                   sec|x - --|
             1                        \    2 /
---------------------------- - ---------------
/          1      \                   sec(x)  
|1 - -------------|*sec(2*x)   1 - -----------
|       /      pi\|                   /    pi\
|    sec|2*x - --||                sec|x - --|
\       \      2 //                   \    2 /
−sec⁡(x)sec⁡(x−π2)+1−sec⁡(x)sec⁡(x−π2)+1+1(1−1sec⁡(2x−π2))sec⁡(2x)- \frac{\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + 1}{- \frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + 1} + \frac{1}{\left(1 - \frac{1}{\sec{\left(2 x - \frac{\pi}{2} \right)}}\right) \sec{\left(2 x \right)}} 
                                2              
  1 + tan(x)             1 - tan (x)           
- ---------- - --------------------------------
  1 - tan(x)   /       2   \ /       2*tan(x) \
               \1 + tan (x)/*|-1 + -----------|
                             |            2   |
                             \     1 + tan (x)/
−tan⁡(x)+11−tan⁡(x)−1−tan⁡2(x)(−1+2tan⁡(x)tan⁡2(x)+1)(tan⁡2(x)+1)- \frac{\tan{\left(x \right)} + 1}{1 - \tan{\left(x \right)}} - \frac{1 - \tan^{2}{\left(x \right)}}{\left(-1 + \frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}\right) \left(\tan^{2}{\left(x \right)} + 1\right)} 
                                2             
  1 + tan(x)             1 - tan (x)          
- ---------- + -------------------------------
  1 - tan(x)   /       2   \ /      2*tan(x) \
               \1 + tan (x)/*|1 - -----------|
                             |           2   |
                             \    1 + tan (x)/
−tan⁡(x)+11−tan⁡(x)+1−tan⁡2(x)(1−2tan⁡(x)tan⁡2(x)+1)(tan⁡2(x)+1)- \frac{\tan{\left(x \right)} + 1}{1 - \tan{\left(x \right)}} + \frac{1 - \tan^{2}{\left(x \right)}}{\left(1 - \frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}\right) \left(\tan^{2}{\left(x \right)} + 1\right)} 
        1                                     
  1 + ------                     2            
      cot(x)             -1 + cot (x)         
- ---------- + -------------------------------
        1      /       2   \ /      2*cot(x) \
  1 - ------   \1 + cot (x)/*|1 - -----------|
      cot(x)                 |           2   |
                             \    1 + cot (x)/
−1+1cot⁡(x)1−1cot⁡(x)+cot⁡2(x)−1(1−2cot⁡(x)cot⁡2(x)+1)(cot⁡2(x)+1)- \frac{1 + \frac{1}{\cot{\left(x \right)}}}{1 - \frac{1}{\cot{\left(x \right)}}} + \frac{\cot^{2}{\left(x \right)} - 1}{\left(1 - \frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right) \left(\cot^{2}{\left(x \right)} + 1\right)} 
                         2   
   /pi      \       2*sin (x)
sin|-- + 2*x|   1 + ---------
   \2       /        sin(2*x)
------------- - -------------
 1 - sin(2*x)            2   
                    2*sin (x)
                1 - ---------
                     sin(2*x)
−2sin⁡2(x)sin⁡(2x)+1−2sin⁡2(x)sin⁡(2x)+1+sin⁡(2x+π2)1−sin⁡(2x)- \frac{\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 1}{- \frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 1} + \frac{\sin{\left(2 x + \frac{\pi}{2} \right)}}{1 - \sin{\left(2 x \right)}} 
sin(pi/2 + 2*x)/(1 - sin(2*x)) - (1 + 2*sin(x)^2/sin(2*x))/(1 - 2*sin(x)^2/sin(2*x))
Abrimos la expresión [src] 
                                                            2        
      1                 1              tan(x)          2*cos (x)     
- ---------- - ------------------- - ---------- + -------------------
  1 - tan(x)   1 - 2*cos(x)*sin(x)   1 - tan(x)   1 - 2*cos(x)*sin(x)
2cos⁡2(x)−2sin⁡(x)cos⁡(x)+1−1−2sin⁡(x)cos⁡(x)+1−tan⁡(x)1−tan⁡(x)−11−tan⁡(x)\frac{2 \cos^{2}{\left(x \right)}}{- 2 \sin{\left(x \right)} \cos{\left(x \right)} + 1} - \frac{1}{- 2 \sin{\left(x \right)} \cos{\left(x \right)} + 1} - \frac{\tan{\left(x \right)}}{1 - \tan{\left(x \right)}} - \frac{1}{1 - \tan{\left(x \right)}} 
-1/(1 - tan(x)) - 1/(1 - 2*cos(x)*sin(x)) - tan(x)/(1 - tan(x)) + 2*cos(x)^2/(1 - 2*cos(x)*sin(x))
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