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Simplificación de expresiones/ Descomposición en fracciones simples/ tan(x)/(1-tan(x)^(2))+cot(x)/(1-cot(x)^(2))

¿Cómo vas a descomponer esta tan(x)/(1-tan(x)^(2))+cot(x)/(1-cot(x)^(2)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
   tan(x)        cot(x)  
----------- + -----------
       2             2   
1 - tan (x)   1 - cot (x)
cot⁡(x)1−cot⁡2(x)+tan⁡(x)1−tan⁡2(x)\frac{\cot{\left(x \right)}}{1 - \cot^{2}{\left(x \right)}} + \frac{\tan{\left(x \right)}}{1 - \tan^{2}{\left(x \right)}} 
tan(x)/(1 - tan(x)^2) + cot(x)/(1 - cot(x)^2)
Simplificación general [src] 
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Respuesta numérica [src] 
cot(x)/(1.0 - cot(x)^2) + tan(x)/(1.0 - tan(x)^2)
cot(x)/(1.0 - cot(x)^2) + tan(x)/(1.0 - tan(x)^2)
Denominador racional [src] 
/       2   \          /       2   \       
\1 - cot (x)/*tan(x) + \1 - tan (x)/*cot(x)
-------------------------------------------
        /       2   \ /       2   \        
        \1 - cot (x)/*\1 - tan (x)/
(1−tan⁡2(x))cot⁡(x)+(1−cot⁡2(x))tan⁡(x)(1−tan⁡2(x))(1−cot⁡2(x))\frac{\left(1 - \tan^{2}{\left(x \right)}\right) \cot{\left(x \right)} + \left(1 - \cot^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\left(1 - \tan^{2}{\left(x \right)}\right) \left(1 - \cot^{2}{\left(x \right)}\right)} 
((1 - cot(x)^2)*tan(x) + (1 - tan(x)^2)*cot(x))/((1 - cot(x)^2)*(1 - tan(x)^2))
Unión de expresiones racionales [src] 
/       2   \          /       2   \       
\1 - cot (x)/*tan(x) + \1 - tan (x)/*cot(x)
-------------------------------------------
        /       2   \ /       2   \        
        \1 - cot (x)/*\1 - tan (x)/
(1−tan⁡2(x))cot⁡(x)+(1−cot⁡2(x))tan⁡(x)(1−tan⁡2(x))(1−cot⁡2(x))\frac{\left(1 - \tan^{2}{\left(x \right)}\right) \cot{\left(x \right)} + \left(1 - \cot^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\left(1 - \tan^{2}{\left(x \right)}\right) \left(1 - \cot^{2}{\left(x \right)}\right)} 
((1 - cot(x)^2)*tan(x) + (1 - tan(x)^2)*cot(x))/((1 - cot(x)^2)*(1 - tan(x)^2))
Combinatoria [src] 
       -(-1 + cot(x)*tan(x))*(cot(x) + tan(x))       
-----------------------------------------------------
(1 + cot(x))*(1 + tan(x))*(-1 + cot(x))*(-1 + tan(x))
−(tan⁡(x)cot⁡(x)−1)(tan⁡(x)+cot⁡(x))(tan⁡(x)−1)(tan⁡(x)+1)(cot⁡(x)−1)(cot⁡(x)+1)- \frac{\left(\tan{\left(x \right)} \cot{\left(x \right)} - 1\right) \left(\tan{\left(x \right)} + \cot{\left(x \right)}\right)}{\left(\tan{\left(x \right)} - 1\right) \left(\tan{\left(x \right)} + 1\right) \left(\cot{\left(x \right)} - 1\right) \left(\cot{\left(x \right)} + 1\right)} 
-(-1 + cot(x)*tan(x))*(cot(x) + tan(x))/((1 + cot(x))*(1 + tan(x))*(-1 + cot(x))*(-1 + tan(x)))
Parte trigonométrica [src] 
     2                    2           
  sin (2*x)          2*sin (x)        
- --------- + ------------------------
   sin(4*x)   /         4   \         
              |    4*sin (x)|         
              |1 - ---------|*sin(2*x)
              |       2     |         
              \    sin (2*x)/
−sin⁡2(2x)sin⁡(4x)+2sin⁡2(x)(−4sin⁡4(x)sin⁡2(2x)+1)sin⁡(2x)- \frac{\sin^{2}{\left(2 x \right)}}{\sin{\left(4 x \right)}} + \frac{2 \sin^{2}{\left(x \right)}}{\left(- \frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1\right) \sin{\left(2 x \right)}} 
                                        2           
         sin(2*x)                  2*sin (x)        
------------------------- + ------------------------
  /       2     \           /         4   \         
  |    sin (2*x)|    2      |    4*sin (x)|         
2*|1 - ---------|*sin (x)   |1 - ---------|*sin(2*x)
  |         4   |           |       2     |         
  \    4*sin (x)/           \    sin (2*x)/
2sin⁡2(x)(−4sin⁡4(x)sin⁡2(2x)+1)sin⁡(2x)+sin⁡(2x)2(1−sin⁡2(2x)4sin⁡4(x))sin⁡2(x)\frac{2 \sin^{2}{\left(x \right)}}{\left(- \frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1\right) \sin{\left(2 x \right)}} + \frac{\sin{\left(2 x \right)}}{2 \left(1 - \frac{\sin^{2}{\left(2 x \right)}}{4 \sin^{4}{\left(x \right)}}\right) \sin^{2}{\left(x \right)}} 
                                 /      pi\        
                            2*sec|2*x - --|        
      sec(2*x)                   \      2 /        
- --------------- + -------------------------------
       /      pi\   /         2      \             
  2*sec|2*x - --|   |      sec (x)   |    2/    pi\
       \      2 /   |1 - ------------|*sec |x - --|
                    |       2/    pi\|     \    2 /
                    |    sec |x - --||             
                    \        \    2 //
−sec⁡(2x)2sec⁡(2x−π2)+2sec⁡(2x−π2)(−sec⁡2(x)sec⁡2(x−π2)+1)sec⁡2(x−π2)- \frac{\sec{\left(2 x \right)}}{2 \sec{\left(2 x - \frac{\pi}{2} \right)}} + \frac{2 \sec{\left(2 x - \frac{\pi}{2} \right)}}{\left(- \frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right) \sec^{2}{\left(x - \frac{\pi}{2} \right)}} 
          /    pi\                                        
       sec|x - --|                                        
          \    2 /                      sec(x)            
------------------------- + ------------------------------
/       2/    pi\\          /         2      \            
|    sec |x - --||          |      sec (x)   |    /    pi\
|        \    2 /|          |1 - ------------|*sec|x - --|
|1 - ------------|*sec(x)   |       2/    pi\|    \    2 /
|         2      |          |    sec |x - --||            
\      sec (x)   /          \        \    2 //
sec⁡(x)(−sec⁡2(x)sec⁡2(x−π2)+1)sec⁡(x−π2)+sec⁡(x−π2)(1−sec⁡2(x−π2)sec⁡2(x))sec⁡(x)\frac{\sec{\left(x \right)}}{\left(- \frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(x - \frac{\pi}{2} \right)}} + \frac{\sec{\left(x - \frac{\pi}{2} \right)}}{\left(1 - \frac{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(x \right)}}\right) \sec{\left(x \right)}} 
         1                tan(x)  
-------------------- + -----------
/       1   \                 2   
|1 - -------|*tan(x)   1 - tan (x)
|       2   |                     
\    tan (x)/
tan⁡(x)1−tan⁡2(x)+1(1−1tan⁡2(x))tan⁡(x)\frac{\tan{\left(x \right)}}{1 - \tan^{2}{\left(x \right)}} + \frac{1}{\left(1 - \frac{1}{\tan^{2}{\left(x \right)}}\right) \tan{\left(x \right)}} 
                          2/x\ /       2   \      
                     4*cot |-|*\1 + cot (x)/      
      1                    \2/                    
- ---------- + -----------------------------------
  2*cot(2*x)                2                     
               /       2/x\\  /       1   \       
               |1 + cot |-|| *|1 - -------|*cot(x)
               \        \2//  |       2   |       
                              \    cot (x)/
−12cot⁡(2x)+4(cot⁡2(x)+1)cot⁡2(x2)(1−1cot⁡2(x))(cot⁡2(x2)+1)2cot⁡(x)- \frac{1}{2 \cot{\left(2 x \right)}} + \frac{4 \left(\cot^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(\frac{x}{2} \right)}}{\left(1 - \frac{1}{\cot^{2}{\left(x \right)}}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \cot{\left(x \right)}} 
       sec(x)                 csc(x)       
-------------------- + --------------------
/       2   \          /       2   \       
|    sec (x)|          |    csc (x)|       
|1 - -------|*csc(x)   |1 - -------|*sec(x)
|       2   |          |       2   |       
\    csc (x)/          \    sec (x)/
csc⁡(x)(−csc⁡2(x)sec⁡2(x)+1)sec⁡(x)+sec⁡(x)(1−sec⁡2(x)csc⁡2(x))csc⁡(x)\frac{\csc{\left(x \right)}}{\left(- \frac{\csc^{2}{\left(x \right)}}{\sec^{2}{\left(x \right)}} + 1\right) \sec{\left(x \right)}} + \frac{\sec{\left(x \right)}}{\left(1 - \frac{\sec^{2}{\left(x \right)}}{\csc^{2}{\left(x \right)}}\right) \csc{\left(x \right)}} 
                        2          
  tan(2*x)         2*sin (x)       
- -------- + ----------------------
     2       /       2   \         
             \1 - tan (x)/*sin(2*x)
−tan⁡(2x)2+2sin⁡2(x)(1−tan⁡2(x))sin⁡(2x)- \frac{\tan{\left(2 x \right)}}{2} + \frac{2 \sin^{2}{\left(x \right)}}{\left(1 - \tan^{2}{\left(x \right)}\right) \sin{\left(2 x \right)}} 
                  2            
  tan(2*x)   2*sin (x)*csc(2*x)
- -------- + ------------------
     2                 2       
                1 - tan (x)
−tan⁡(2x)2+2sin⁡2(x)csc⁡(2x)1−tan⁡2(x)- \frac{\tan{\left(2 x \right)}}{2} + \frac{2 \sin^{2}{\left(x \right)} \csc{\left(2 x \right)}}{1 - \tan^{2}{\left(x \right)}} 
0
00 
          /pi    \                                        
       csc|-- - x|                                        
          \2     /                      csc(x)            
------------------------- + ------------------------------
/       2/pi    \\          /         2      \            
|    csc |-- - x||          |      csc (x)   |    /pi    \
|        \2     /|          |1 - ------------|*csc|-- - x|
|1 - ------------|*csc(x)   |       2/pi    \|    \2     /
|         2      |          |    csc |-- - x||            
\      csc (x)   /          \        \2     //
csc⁡(x)(−csc⁡2(x)csc⁡2(−x+π2)+1)csc⁡(−x+π2)+csc⁡(−x+π2)(1−csc⁡2(−x+π2)csc⁡2(x))csc⁡(x)\frac{\csc{\left(x \right)}}{\left(- \frac{\csc^{2}{\left(x \right)}}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\left(1 - \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right) \csc{\left(x \right)}} 
     /pi      \                             
  csc|-- - 2*x|                             
     \2       /           2*csc(2*x)        
- ------------- + --------------------------
    2*csc(2*x)    /       2/pi    \\        
                  |    csc |-- - x||        
                  |        \2     /|    2   
                  |1 - ------------|*csc (x)
                  |         2      |        
                  \      csc (x)   /
−csc⁡(−2x+π2)2csc⁡(2x)+2csc⁡(2x)(1−csc⁡2(−x+π2)csc⁡2(x))csc⁡2(x)- \frac{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}{2 \csc{\left(2 x \right)}} + \frac{2 \csc{\left(2 x \right)}}{\left(1 - \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right) \csc^{2}{\left(x \right)}} 
          /    pi\                                        
       cos|x - --|                                        
          \    2 /                      cos(x)            
------------------------- + ------------------------------
/       2/    pi\\          /         2      \            
|    cos |x - --||          |      cos (x)   |    /    pi\
|        \    2 /|          |1 - ------------|*cos|x - --|
|1 - ------------|*cos(x)   |       2/    pi\|    \    2 /
|         2      |          |    cos |x - --||            
\      cos (x)   /          \        \    2 //
cos⁡(x)(−cos⁡2(x)cos⁡2(x−π2)+1)cos⁡(x−π2)+cos⁡(x−π2)(1−cos⁡2(x−π2)cos⁡2(x))cos⁡(x)\frac{\cos{\left(x \right)}}{\left(- \frac{\cos^{2}{\left(x \right)}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(x - \frac{\pi}{2} \right)}} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\left(1 - \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right) \cos{\left(x \right)}} 
         1                cot(x)  
-------------------- + -----------
/       1   \                 2   
|1 - -------|*cot(x)   1 - cot (x)
|       2   |                     
\    cot (x)/
cot⁡(x)1−cot⁡2(x)+1(1−1cot⁡2(x))cot⁡(x)\frac{\cot{\left(x \right)}}{1 - \cot^{2}{\left(x \right)}} + \frac{1}{\left(1 - \frac{1}{\cot^{2}{\left(x \right)}}\right) \cot{\left(x \right)}} 
     /      pi\                 2/    pi\         
  cos|2*x - --|            2*cos |x - --|         
     \      2 /                  \    2 /         
- ------------- + --------------------------------
    2*cos(2*x)    /       2/    pi\\              
                  |    cos |x - --||              
                  |        \    2 /|    /      pi\
                  |1 - ------------|*cos|2*x - --|
                  |         2      |    \      2 /
                  \      cos (x)   /
−cos⁡(2x−π2)2cos⁡(2x)+2cos⁡2(x−π2)(1−cos⁡2(x−π2)cos⁡2(x))cos⁡(2x−π2)- \frac{\cos{\left(2 x - \frac{\pi}{2} \right)}}{2 \cos{\left(2 x \right)}} + \frac{2 \cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\left(1 - \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right) \cos{\left(2 x - \frac{\pi}{2} \right)}} 
       sin(x)                 cos(x)       
-------------------- + --------------------
/       2   \          /       2   \       
|    sin (x)|          |    cos (x)|       
|1 - -------|*cos(x)   |1 - -------|*sin(x)
|       2   |          |       2   |       
\    cos (x)/          \    sin (x)/
sin⁡(x)(−sin⁡2(x)cos⁡2(x)+1)cos⁡(x)+cos⁡(x)(1−cos⁡2(x)sin⁡2(x))sin⁡(x)\frac{\sin{\left(x \right)}}{\left(- \frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \cos{\left(x \right)}} + \frac{\cos{\left(x \right)}}{\left(1 - \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \sin{\left(x \right)}} 
                        2/x\ /       2   \      
                   4*tan |-|*\1 + tan (x)/      
  tan(2*x)               \2/                    
- -------- + -----------------------------------
     2                    2                     
             /       2/x\\  /       2   \       
             |1 + tan |-|| *\1 - tan (x)/*tan(x)
             \        \2//
−tan⁡(2x)2+4(tan⁡2(x)+1)tan⁡2(x2)(1−tan⁡2(x))(tan⁡2(x2)+1)2tan⁡(x)- \frac{\tan{\left(2 x \right)}}{2} + \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(\frac{x}{2} \right)}}{\left(1 - \tan^{2}{\left(x \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \tan{\left(x \right)}} 
-tan(2*x)/2 + 4*tan(x/2)^2*(1 + tan(x)^2)/((1 + tan(x/2)^2)^2*(1 - tan(x)^2)*tan(x))
Potencias [src] 
                          /   I*x    -I*x\          
   cot(x)               I*\- e    + e    /          
----------- + --------------------------------------
       2      /                    2\               
1 - cot (x)   |    /   I*x    -I*x\ |               
              |    \- e    + e    / | / I*x    -I*x\
              |1 + -----------------|*\e    + e    /
              |                   2 |               
              |     / I*x    -I*x\  |               
              \     \e    + e    /  /
i(−eix+e−ix)((−eix+e−ix)2(eix+e−ix)2+1)(eix+e−ix)+cot⁡(x)1−cot⁡2(x)\frac{i \left(- e^{i x} + e^{- i x}\right)}{\left(\frac{\left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} + 1\right) \left(e^{i x} + e^{- i x}\right)} + \frac{\cot{\left(x \right)}}{1 - \cot^{2}{\left(x \right)}} 
cot(x)/(1 - cot(x)^2) + i*(-exp(i*x) + exp(-i*x))/((1 + (-exp(i*x) + exp(-i*x))^2/(exp(i*x) + exp(-i*x))^2)*(exp(i*x) + exp(-i*x)))
Denominador común [src] 
 /                      2                2          \ 
-\-cot(x) - tan(x) + cot (x)*tan(x) + tan (x)*cot(x)/ 
------------------------------------------------------
              2         2         2       2           
       1 - cot (x) - tan (x) + cot (x)*tan (x)
−tan⁡2(x)cot⁡(x)+tan⁡(x)cot⁡2(x)−tan⁡(x)−cot⁡(x)tan⁡2(x)cot⁡2(x)−tan⁡2(x)−cot⁡2(x)+1- \frac{\tan^{2}{\left(x \right)} \cot{\left(x \right)} + \tan{\left(x \right)} \cot^{2}{\left(x \right)} - \tan{\left(x \right)} - \cot{\left(x \right)}}{\tan^{2}{\left(x \right)} \cot^{2}{\left(x \right)} - \tan^{2}{\left(x \right)} - \cot^{2}{\left(x \right)} + 1} 
-(-cot(x) - tan(x) + cot(x)^2*tan(x) + tan(x)^2*cot(x))/(1 - cot(x)^2 - tan(x)^2 + cot(x)^2*tan(x)^2)
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