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Simplificación de expresiones/ Descomposición en fracciones simples/ tg^2(x)*ctg^2(x)-(1-sin^2(5x))/(cos^2(5x)-1)

¿Cómo vas a descomponer esta tg^2(x)*ctg^2(x)-(1-sin^2(5x))/(cos^2(5x)-1) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
                         2     
   2       2      1 - sin (5*x)
tan (x)*cot (x) - -------------
                     2         
                  cos (5*x) - 1
−1−sin⁡2(5x)cos⁡2(5x)−1+tan⁡2(x)cot⁡2(x)- \frac{1 - \sin^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)} - 1} + \tan^{2}{\left(x \right)} \cot^{2}{\left(x \right)} 
tan(x)^2*cot(x)^2 - (1 - sin(5*x)^2)/(cos(5*x)^2 - 1)
Simplificación general [src] 
    1    
---------
   2     
sin (5*x)
1sin⁡2(5x)\frac{1}{\sin^{2}{\left(5 x \right)}} 
sin(5*x)^(-2)
Respuesta numérica [src] 
cot(x)^2*tan(x)^2 - (1.0 - sin(5*x)^2)/(-1.0 + cos(5*x)^2)
cot(x)^2*tan(x)^2 - (1.0 - sin(5*x)^2)/(-1.0 + cos(5*x)^2)
Unión de expresiones racionales [src] 
        2           2       2    /        2     \
-1 + sin (5*x) + cot (x)*tan (x)*\-1 + cos (5*x)/
-------------------------------------------------
                          2                      
                  -1 + cos (5*x)
(cos⁡2(5x)−1)tan⁡2(x)cot⁡2(x)+sin⁡2(5x)−1cos⁡2(5x)−1\frac{\left(\cos^{2}{\left(5 x \right)} - 1\right) \tan^{2}{\left(x \right)} \cot^{2}{\left(x \right)} + \sin^{2}{\left(5 x \right)} - 1}{\cos^{2}{\left(5 x \right)} - 1} 
(-1 + sin(5*x)^2 + cot(x)^2*tan(x)^2*(-1 + cos(5*x)^2))/(-1 + cos(5*x)^2)
Denominador racional [src] 
        2           2       2    /        2     \
-1 + sin (5*x) + cot (x)*tan (x)*\-1 + cos (5*x)/
-------------------------------------------------
                          2                      
                  -1 + cos (5*x)
(cos⁡2(5x)−1)tan⁡2(x)cot⁡2(x)+sin⁡2(5x)−1cos⁡2(5x)−1\frac{\left(\cos^{2}{\left(5 x \right)} - 1\right) \tan^{2}{\left(x \right)} \cot^{2}{\left(x \right)} + \sin^{2}{\left(5 x \right)} - 1}{\cos^{2}{\left(5 x \right)} - 1} 
(-1 + sin(5*x)^2 + cot(x)^2*tan(x)^2*(-1 + cos(5*x)^2))/(-1 + cos(5*x)^2)
Denominador común [src] 
        2                       
-1 + sin (5*x)      2       2   
-------------- + cot (x)*tan (x)
        2                       
-1 + cos (5*x)
sin⁡2(5x)−1cos⁡2(5x)−1+tan⁡2(x)cot⁡2(x)\frac{\sin^{2}{\left(5 x \right)} - 1}{\cos^{2}{\left(5 x \right)} - 1} + \tan^{2}{\left(x \right)} \cot^{2}{\left(x \right)} 
(-1 + sin(5*x)^2)/(-1 + cos(5*x)^2) + cot(x)^2*tan(x)^2
Potencias [src] 
                          2                            
      /   -5*I*x    5*I*x\                             
      \- e       + e     /                    2        
  1 + ---------------------   /   I*x    -I*x\     2   
                4             \- e    + e    / *cot (x)
- ------------------------- - -------------------------
                          2                      2     
        / -5*I*x    5*I*x\         / I*x    -I*x\      
        |e         e     |         \e    + e    /      
   -1 + |------- + ------|                             
        \   2        2   /
−(−eix+e−ix)2cot⁡2(x)(eix+e−ix)2−(e5ix−e−5ix)24+1(e5ix2+e−5ix2)2−1- \frac{\left(- e^{i x} + e^{- i x}\right)^{2} \cot^{2}{\left(x \right)}}{\left(e^{i x} + e^{- i x}\right)^{2}} - \frac{\frac{\left(e^{5 i x} - e^{- 5 i x}\right)^{2}}{4} + 1}{\left(\frac{e^{5 i x}}{2} + \frac{e^{- 5 i x}}{2}\right)^{2} - 1} 
        2                       
-1 + sin (5*x)      2       2   
-------------- + cot (x)*tan (x)
        2                       
-1 + cos (5*x)
sin⁡2(5x)−1cos⁡2(5x)−1+tan⁡2(x)cot⁡2(x)\frac{\sin^{2}{\left(5 x \right)} - 1}{\cos^{2}{\left(5 x \right)} - 1} + \tan^{2}{\left(x \right)} \cot^{2}{\left(x \right)} 
(-1 + sin(5*x)^2)/(-1 + cos(5*x)^2) + cot(x)^2*tan(x)^2
Abrimos la expresión [src] 
                                                                                                                                    8                                                                          4                                                                          2                                                                          10                                                                         6                                  
                                     1                                          2       2                                    640*sin (x)                                                                200*sin (x)                                                                 25*sin (x)                                                                256*sin  (x)                                                               560*sin (x)                               
- ------------------------------------------------------------------------ + tan (x)*cot (x) - ------------------------------------------------------------------------ - ------------------------------------------------------------------------ + ------------------------------------------------------------------------ + ------------------------------------------------------------------------ + ------------------------------------------------------------------------
              8             4            2             10             6                                    8             4            2             10             6                  8             4            2             10             6                  8             4            2             10             6                  8             4            2             10             6                  8             4            2             10             6   
  -1 - 640*cos (x) - 200*cos (x) + 25*cos (x) + 256*cos  (x) + 560*cos (x)                     -1 - 640*cos (x) - 200*cos (x) + 25*cos (x) + 256*cos  (x) + 560*cos (x)   -1 - 640*cos (x) - 200*cos (x) + 25*cos (x) + 256*cos  (x) + 560*cos (x)   -1 - 640*cos (x) - 200*cos (x) + 25*cos (x) + 256*cos  (x) + 560*cos (x)   -1 - 640*cos (x) - 200*cos (x) + 25*cos (x) + 256*cos  (x) + 560*cos (x)   -1 - 640*cos (x) - 200*cos (x) + 25*cos (x) + 256*cos  (x) + 560*cos (x)
tan⁡2(x)cot⁡2(x)+256sin⁡10(x)256cos⁡10(x)−640cos⁡8(x)+560cos⁡6(x)−200cos⁡4(x)+25cos⁡2(x)−1−640sin⁡8(x)256cos⁡10(x)−640cos⁡8(x)+560cos⁡6(x)−200cos⁡4(x)+25cos⁡2(x)−1+560sin⁡6(x)256cos⁡10(x)−640cos⁡8(x)+560cos⁡6(x)−200cos⁡4(x)+25cos⁡2(x)−1−200sin⁡4(x)256cos⁡10(x)−640cos⁡8(x)+560cos⁡6(x)−200cos⁡4(x)+25cos⁡2(x)−1+25sin⁡2(x)256cos⁡10(x)−640cos⁡8(x)+560cos⁡6(x)−200cos⁡4(x)+25cos⁡2(x)−1−1256cos⁡10(x)−640cos⁡8(x)+560cos⁡6(x)−200cos⁡4(x)+25cos⁡2(x)−1\tan^{2}{\left(x \right)} \cot^{2}{\left(x \right)} + \frac{256 \sin^{10}{\left(x \right)}}{256 \cos^{10}{\left(x \right)} - 640 \cos^{8}{\left(x \right)} + 560 \cos^{6}{\left(x \right)} - 200 \cos^{4}{\left(x \right)} + 25 \cos^{2}{\left(x \right)} - 1} - \frac{640 \sin^{8}{\left(x \right)}}{256 \cos^{10}{\left(x \right)} - 640 \cos^{8}{\left(x \right)} + 560 \cos^{6}{\left(x \right)} - 200 \cos^{4}{\left(x \right)} + 25 \cos^{2}{\left(x \right)} - 1} + \frac{560 \sin^{6}{\left(x \right)}}{256 \cos^{10}{\left(x \right)} - 640 \cos^{8}{\left(x \right)} + 560 \cos^{6}{\left(x \right)} - 200 \cos^{4}{\left(x \right)} + 25 \cos^{2}{\left(x \right)} - 1} - \frac{200 \sin^{4}{\left(x \right)}}{256 \cos^{10}{\left(x \right)} - 640 \cos^{8}{\left(x \right)} + 560 \cos^{6}{\left(x \right)} - 200 \cos^{4}{\left(x \right)} + 25 \cos^{2}{\left(x \right)} - 1} + \frac{25 \sin^{2}{\left(x \right)}}{256 \cos^{10}{\left(x \right)} - 640 \cos^{8}{\left(x \right)} + 560 \cos^{6}{\left(x \right)} - 200 \cos^{4}{\left(x \right)} + 25 \cos^{2}{\left(x \right)} - 1} - \frac{1}{256 \cos^{10}{\left(x \right)} - 640 \cos^{8}{\left(x \right)} + 560 \cos^{6}{\left(x \right)} - 200 \cos^{4}{\left(x \right)} + 25 \cos^{2}{\left(x \right)} - 1} 
-1/(-1 - 640*cos(x)^8 - 200*cos(x)^4 + 25*cos(x)^2 + 256*cos(x)^10 + 560*cos(x)^6) + tan(x)^2*cot(x)^2 - 640*sin(x)^8/(-1 - 640*cos(x)^8 - 200*cos(x)^4 + 25*cos(x)^2 + 256*cos(x)^10 + 560*cos(x)^6) - 200*sin(x)^4/(-1 - 640*cos(x)^8 - 200*cos(x)^4 + 25*cos(x)^2 + 256*cos(x)^10 + 560*cos(x)^6) + 25*sin(x)^2/(-1 - 640*cos(x)^8 - 200*cos(x)^4 + 25*cos(x)^2 + 256*cos(x)^10 + 560*cos(x)^6) + 256*sin(x)^10/(-1 - 640*cos(x)^8 - 200*cos(x)^4 + 25*cos(x)^2 + 256*cos(x)^10 + 560*cos(x)^6) + 560*sin(x)^6/(-1 - 640*cos(x)^8 - 200*cos(x)^4 + 25*cos(x)^2 + 256*cos(x)^10 + 560*cos(x)^6)
Parte trigonométrica [src] 
                2/5*x\   
           4*tan |---|   
                 \ 2 /   
     1 - ----------------
                        2
         /       2/5*x\\ 
         |1 + tan |---|| 
         \        \ 2 // 
1 - ---------------------
                        2
         /       2/5*x\\ 
         |1 - tan |---|| 
         \        \ 2 // 
    -1 + ----------------
                        2
         /       2/5*x\\ 
         |1 + tan |---|| 
         \        \ 2 //
−1−4tan⁡2(5x2)(tan⁡2(5x2)+1)2(1−tan⁡2(5x2))2(tan⁡2(5x2)+1)2−1+1- \frac{1 - \frac{4 \tan^{2}{\left(\frac{5 x}{2} \right)}}{\left(\tan^{2}{\left(\frac{5 x}{2} \right)} + 1\right)^{2}}}{\frac{\left(1 - \tan^{2}{\left(\frac{5 x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{5 x}{2} \right)} + 1\right)^{2}} - 1} + 1 
   2     
csc (5*x)
csc⁡2(5x)\csc^{2}{\left(5 x \right)} 
   2/      pi\
sec |5*x - --|
    \      2 /
sec⁡2(5x−π2)\sec^{2}{\left(5 x - \frac{\pi}{2} \right)} 
      2      
-------------
1 - cos(10*x)
21−cos⁡(10x)\frac{2}{1 - \cos{\left(10 x \right)}} 
           2/      pi\
    1 - cos |5*x - --|
            \      2 /
1 - ------------------
              2       
      -1 + cos (5*x)
−1−cos⁡2(5x−π2)cos⁡2(5x)−1+1- \frac{1 - \cos^{2}{\left(5 x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(5 x \right)} - 1} + 1 
      1       
--------------
   2/      pi\
cos |5*x - --|
    \      2 /
1cos⁡2(5x−π2)\frac{1}{\cos^{2}{\left(5 x - \frac{\pi}{2} \right)}} 
                2/5*x\    
           4*cot |---|    
                 \ 2 /    
     1 - ---------------- 
                        2 
         /       2/5*x\\  
         |1 + cot |---||  
         \        \ 2 //  
1 - ----------------------
                         2
         /        2/5*x\\ 
         |-1 + cot |---|| 
         \         \ 2 // 
    -1 + -----------------
                         2
          /       2/5*x\\ 
          |1 + cot |---|| 
          \        \ 2 //
−1−4cot⁡2(5x2)(cot⁡2(5x2)+1)2(cot⁡2(5x2)−1)2(cot⁡2(5x2)+1)2−1+1- \frac{1 - \frac{4 \cot^{2}{\left(\frac{5 x}{2} \right)}}{\left(\cot^{2}{\left(\frac{5 x}{2} \right)} + 1\right)^{2}}}{\frac{\left(\cot^{2}{\left(\frac{5 x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{5 x}{2} \right)} + 1\right)^{2}} - 1} + 1 
            1     
    1 - --------- 
           2      
        csc (5*x) 
1 - --------------
             1    
    -1 + ---------
            2     
         sec (5*x)
1−1−1csc⁡2(5x)−1+1sec⁡2(5x)1 - \frac{1 - \frac{1}{\csc^{2}{\left(5 x \right)}}}{-1 + \frac{1}{\sec^{2}{\left(5 x \right)}}} 
              1       
    1 - --------------
           2/      pi\
        sec |5*x - --|
            \      2 /
1 - ------------------
               1      
      -1 + ---------  
              2       
           sec (5*x)
1−1−1sec⁡2(5x−π2)−1+1sec⁡2(5x)1 - \frac{1 - \frac{1}{\sec^{2}{\left(5 x - \frac{\pi}{2} \right)}}}{-1 + \frac{1}{\sec^{2}{\left(5 x \right)}}} 
               2
/       2/5*x\\ 
|1 + cot |---|| 
\        \ 2 // 
----------------
       2/5*x\   
  4*cot |---|   
        \ 2 /
(cot⁡2(5x2)+1)24cot⁡2(5x2)\frac{\left(\cot^{2}{\left(\frac{5 x}{2} \right)} + 1\right)^{2}}{4 \cot^{2}{\left(\frac{5 x}{2} \right)}} 
           2      
    1 - sin (5*x) 
1 - --------------
            2     
    -1 + cos (5*x)
−1−sin⁡2(5x)cos⁡2(5x)−1+1- \frac{1 - \sin^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)} - 1} + 1 
               2
/       2/5*x\\ 
|1 + tan |---|| 
\        \ 2 // 
----------------
       2/5*x\   
  4*tan |---|   
        \ 2 /
(tan⁡2(5x2)+1)24tan⁡2(5x2)\frac{\left(\tan^{2}{\left(\frac{5 x}{2} \right)} + 1\right)^{2}}{4 \tan^{2}{\left(\frac{5 x}{2} \right)}} 
               1       
       1 - ---------   
              2        
           csc (5*x)   
1 - -------------------
               1       
    -1 + --------------
            2/pi      \
         csc |-- - 5*x|
             \2       /
1−1−1csc⁡2(5x)−1+1csc⁡2(−5x+π2)1 - \frac{1 - \frac{1}{\csc^{2}{\left(5 x \right)}}}{-1 + \frac{1}{\csc^{2}{\left(- 5 x + \frac{\pi}{2} \right)}}} 
              2        
       1 - sin (5*x)   
1 - -------------------
            2/pi      \
    -1 + sin |-- + 5*x|
             \2       /
−1−sin⁡2(5x)sin⁡2(5x+π2)−1+1- \frac{1 - \sin^{2}{\left(5 x \right)}}{\sin^{2}{\left(5 x + \frac{\pi}{2} \right)} - 1} + 1 
    1    
---------
   2     
sin (5*x)
1sin⁡2(5x)\frac{1}{\sin^{2}{\left(5 x \right)}} 
sin(5*x)^(-2)
Combinatoria [src] 
        2           2       2         2         2       2   
-1 + sin (5*x) - cot (x)*tan (x) + cos (5*x)*cot (x)*tan (x)
------------------------------------------------------------
               (1 + cos(5*x))*(-1 + cos(5*x))
sin⁡2(5x)+cos⁡2(5x)tan⁡2(x)cot⁡2(x)−tan⁡2(x)cot⁡2(x)−1(cos⁡(5x)−1)(cos⁡(5x)+1)\frac{\sin^{2}{\left(5 x \right)} + \cos^{2}{\left(5 x \right)} \tan^{2}{\left(x \right)} \cot^{2}{\left(x \right)} - \tan^{2}{\left(x \right)} \cot^{2}{\left(x \right)} - 1}{\left(\cos{\left(5 x \right)} - 1\right) \left(\cos{\left(5 x \right)} + 1\right)} 
(-1 + sin(5*x)^2 - cot(x)^2*tan(x)^2 + cos(5*x)^2*cot(x)^2*tan(x)^2)/((1 + cos(5*x))*(-1 + cos(5*x)))
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