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Simplificación de expresiones/ Descomposición en fracciones simples/ tan(2*x)/(tan(4*x)-tan(2*x))

¿Cómo vas a descomponer esta tan(2*x)/(tan(4*x)-tan(2*x)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
      tan(2*x)     
-------------------
tan(4*x) - tan(2*x)
$$\frac{\tan{\left(2 x \right)}}{- \tan{\left(2 x \right)} + \tan{\left(4 x \right)}}$$ 
tan(2*x)/(tan(4*x) - tan(2*x))
Simplificación general [src] 
     -tan(2*x)      
--------------------
-tan(4*x) + tan(2*x)
$$- \frac{\tan{\left(2 x \right)}}{\tan{\left(2 x \right)} - \tan{\left(4 x \right)}}$$ 
-tan(2*x)/(-tan(4*x) + tan(2*x))
Respuesta numérica [src] 
tan(2*x)/(-tan(2*x) + tan(4*x))
tan(2*x)/(-tan(2*x) + tan(4*x))
Combinatoria [src] 
     -tan(2*x)      
--------------------
-tan(4*x) + tan(2*x)
$$- \frac{\tan{\left(2 x \right)}}{\tan{\left(2 x \right)} - \tan{\left(4 x \right)}}$$ 
-tan(2*x)/(-tan(4*x) + tan(2*x))
Denominador común [src] 
           tan(4*x)      
-1 - --------------------
     -tan(4*x) + tan(2*x)
$$-1 - \frac{\tan{\left(4 x \right)}}{\tan{\left(2 x \right)} - \tan{\left(4 x \right)}}$$ 
-1 - tan(4*x)/(-tan(4*x) + tan(2*x))
Abrimos la expresión [src] 
                                                 2*tan(x)                                                
---------------------------------------------------------------------------------------------------------
              3                                 3                   5                                    
         8*tan (x)            2*tan(x)     2*tan (x)           4*tan (x)                  4*tan(x)       
- ----------------------- - ----------- + ----------- + ----------------------- + -----------------------
         4           2             2             2             4           2             4           2   
  1 + tan (x) - 6*tan (x)   1 - tan (x)   1 - tan (x)   1 + tan (x) - 6*tan (x)   1 + tan (x) - 6*tan (x)
$$\frac{2 \tan{\left(x \right)}}{\frac{4 \tan^{5}{\left(x \right)}}{\tan^{4}{\left(x \right)} - 6 \tan^{2}{\left(x \right)} + 1} - \frac{8 \tan^{3}{\left(x \right)}}{\tan^{4}{\left(x \right)} - 6 \tan^{2}{\left(x \right)} + 1} + \frac{4 \tan{\left(x \right)}}{\tan^{4}{\left(x \right)} - 6 \tan^{2}{\left(x \right)} + 1} + \frac{2 \tan^{3}{\left(x \right)}}{1 - \tan^{2}{\left(x \right)}} - \frac{2 \tan{\left(x \right)}}{1 - \tan^{2}{\left(x \right)}}}$$ 
2*tan(x)/(-8*tan(x)^3/(1 + tan(x)^4 - 6*tan(x)^2) - 2*tan(x)/(1 - tan(x)^2) + 2*tan(x)^3/(1 - tan(x)^2) + 4*tan(x)^5/(1 + tan(x)^4 - 6*tan(x)^2) + 4*tan(x)/(1 + tan(x)^4 - 6*tan(x)^2))
Parte trigonométrica [src] 
                      /      pi\                  
                 2*sec|4*x - --|                  
                      \      2 /                  
--------------------------------------------------
/                       /      pi\\               
|                  2*sec|8*x - --||               
|     sec(2*x)          \      2 /|    2/      pi\
|- ------------- + ---------------|*sec |2*x - --|
|     /      pi\       2/      pi\|     \      2 /
|  sec|2*x - --|    sec |4*x - --||               
\     \      2 /        \      2 //
$$\frac{2 \sec{\left(4 x - \frac{\pi}{2} \right)}}{\left(- \frac{\sec{\left(2 x \right)}}{\sec{\left(2 x - \frac{\pi}{2} \right)}} + \frac{2 \sec{\left(8 x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(4 x - \frac{\pi}{2} \right)}}\right) \sec^{2}{\left(2 x - \frac{\pi}{2} \right)}}$$ 
         2           4   
1 - 8*sin (x) + 8*sin (x)
$$8 \sin^{4}{\left(x \right)} - 8 \sin^{2}{\left(x \right)} + 1$$ 
                         2    /       2     \                    
                    4*tan (x)*\1 + tan (2*x)/                    
-----------------------------------------------------------------
             2 /                 2      /       2     \\         
/       2   \  |            4*tan (2*x)*\1 + tan (4*x)/|         
\1 + tan (x)/ *|-tan(2*x) + ---------------------------|*tan(2*x)
               |                            2          |         
               |             /       2     \           |         
               \             \1 + tan (2*x)/ *tan(4*x) /
$$\frac{4 \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan^{2}{\left(x \right)}}{\left(\tan^{2}{\left(x \right)} + 1\right)^{2} \left(- \tan{\left(2 x \right)} + \frac{4 \left(\tan^{2}{\left(4 x \right)} + 1\right) \tan^{2}{\left(2 x \right)}}{\left(\tan^{2}{\left(2 x \right)} + 1\right)^{2} \tan{\left(4 x \right)}}\right) \tan{\left(2 x \right)}}$$ 
           sin(2*x)           
------------------------------
/sin(4*x)   sin(2*x)\         
|-------- - --------|*cos(2*x)
\cos(4*x)   cos(2*x)/
$$\frac{\sin{\left(2 x \right)}}{\left(- \frac{\sin{\left(2 x \right)}}{\cos{\left(2 x \right)}} + \frac{\sin{\left(4 x \right)}}{\cos{\left(4 x \right)}}\right) \cos{\left(2 x \right)}}$$ 
 sec(2*x)*sin(2*x)  
--------------------
-tan(2*x) + tan(4*x)
$$\frac{\sin{\left(2 x \right)} \sec{\left(2 x \right)}}{- \tan{\left(2 x \right)} + \tan{\left(4 x \right)}}$$ 
                 2                  
            2*sin (2*x)             
------------------------------------
/                  2      \         
|             2*sin (4*x) |         
|-tan(2*x) + -------------|*sin(4*x)
|               /      pi\|         
|            cos|8*x - --||         
\               \      2 //
$$\frac{2 \sin^{2}{\left(2 x \right)}}{\left(\frac{2 \sin^{2}{\left(4 x \right)}}{\cos{\left(8 x - \frac{\pi}{2} \right)}} - \tan{\left(2 x \right)}\right) \sin{\left(4 x \right)}}$$ 
                2                 
           2*sin (2*x)            
----------------------------------
/                 2     \         
|            2*sin (4*x)|         
|-tan(2*x) + -----------|*sin(4*x)
\              sin(8*x) /
$$\frac{2 \sin^{2}{\left(2 x \right)}}{\left(\frac{2 \sin^{2}{\left(4 x \right)}}{\sin{\left(8 x \right)}} - \tan{\left(2 x \right)}\right) \sin{\left(4 x \right)}}$$ 
                /      pi\              
             cos|2*x - --|              
                \      2 /              
----------------------------------------
/   /      pi\      /      pi\\         
|cos|4*x - --|   cos|2*x - --||         
|   \      2 /      \      2 /|         
|------------- - -------------|*cos(2*x)
\   cos(4*x)        cos(2*x)  /
$$\frac{\cos{\left(2 x - \frac{\pi}{2} \right)}}{\left(\frac{\cos{\left(4 x - \frac{\pi}{2} \right)}}{\cos{\left(4 x \right)}} - \frac{\cos{\left(2 x - \frac{\pi}{2} \right)}}{\cos{\left(2 x \right)}}\right) \cos{\left(2 x \right)}}$$ 
                  2                   
             2*sin (2*x)              
--------------------------------------
/       2             2     \         
|  2*sin (2*x)   2*sin (4*x)|         
|- ----------- + -----------|*sin(4*x)
\    sin(4*x)      sin(8*x) /
$$\frac{2 \sin^{2}{\left(2 x \right)}}{\left(- \frac{2 \sin^{2}{\left(2 x \right)}}{\sin{\left(4 x \right)}} + \frac{2 \sin^{2}{\left(4 x \right)}}{\sin{\left(8 x \right)}}\right) \sin{\left(4 x \right)}}$$ 
           sec(2*x)           
------------------------------
/sec(4*x)   sec(2*x)\         
|-------- - --------|*csc(2*x)
\csc(4*x)   csc(2*x)/
$$\frac{\sec{\left(2 x \right)}}{\left(\frac{\sec{\left(4 x \right)}}{\csc{\left(4 x \right)}} - \frac{\sec{\left(2 x \right)}}{\csc{\left(2 x \right)}}\right) \csc{\left(2 x \right)}}$$ 
              2*tan(x)              
------------------------------------
/       2   \                       
\1 - tan (x)/*(-tan(2*x) + tan(4*x))
$$\frac{2 \tan{\left(x \right)}}{\left(1 - \tan^{2}{\left(x \right)}\right) \left(- \tan{\left(2 x \right)} + \tan{\left(4 x \right)}\right)}$$ 
                      2/      pi\                 
                 2*cos |2*x - --|                 
                       \      2 /                 
--------------------------------------------------
/     /      pi\        2/      pi\\              
|  cos|2*x - --|   2*cos |4*x - --||              
|     \      2 /         \      2 /|    /      pi\
|- ------------- + ----------------|*cos|4*x - --|
|     cos(2*x)         /      pi\  |    \      2 /
|                   cos|8*x - --|  |              
\                      \      2 /  /
$$\frac{2 \cos^{2}{\left(2 x - \frac{\pi}{2} \right)}}{\left(\frac{2 \cos^{2}{\left(4 x - \frac{\pi}{2} \right)}}{\cos{\left(8 x - \frac{\pi}{2} \right)}} - \frac{\cos{\left(2 x - \frac{\pi}{2} \right)}}{\cos{\left(2 x \right)}}\right) \cos{\left(4 x - \frac{\pi}{2} \right)}}$$ 
               2*csc(4*x)               
----------------------------------------
/     /pi      \             \          
|  csc|-- - 2*x|             |          
|     \2       /   2*csc(8*x)|    2     
|- ------------- + ----------|*csc (2*x)
|     csc(2*x)        2      |          
\                  csc (4*x) /
$$\frac{2 \csc{\left(4 x \right)}}{\left(\frac{2 \csc{\left(8 x \right)}}{\csc^{2}{\left(4 x \right)}} - \frac{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}{\csc{\left(2 x \right)}}\right) \csc^{2}{\left(2 x \right)}}$$ 
              2*cot(x)              
------------------------------------
/        2   \ /   1          1    \
\-1 + cot (x)/*|-------- - --------|
               \cot(4*x)   cot(2*x)/
$$\frac{2 \cot{\left(x \right)}}{\left(\cot^{2}{\left(x \right)} - 1\right) \left(\frac{1}{\cot{\left(4 x \right)}} - \frac{1}{\cot{\left(2 x \right)}}\right)}$$ 
                   sec(2*x)                  
---------------------------------------------
/   sec(4*x)        sec(2*x)  \    /      pi\
|------------- - -------------|*sec|2*x - --|
|   /      pi\      /      pi\|    \      2 /
|sec|4*x - --|   sec|2*x - --||              
\   \      2 /      \      2 //
$$\frac{\sec{\left(2 x \right)}}{\left(- \frac{\sec{\left(2 x \right)}}{\sec{\left(2 x - \frac{\pi}{2} \right)}} + \frac{\sec{\left(4 x \right)}}{\sec{\left(4 x - \frac{\pi}{2} \right)}}\right) \sec{\left(2 x - \frac{\pi}{2} \right)}}$$ 
                       2                        
                  2*sin (2*x)                   
------------------------------------------------
/                 2         /      pi\\         
|-tan(2*x) + 2*sin (4*x)*sec|8*x - --||*sin(4*x)
\                           \      2 //
$$\frac{2 \sin^{2}{\left(2 x \right)}}{\left(2 \sin^{2}{\left(4 x \right)} \sec{\left(8 x - \frac{\pi}{2} \right)} - \tan{\left(2 x \right)}\right) \sin{\left(4 x \right)}}$$ 
                  sin(2*x)                 
-------------------------------------------
/       2             2     \              
|  2*sin (2*x)   2*sin (4*x)|    /pi      \
|- ----------- + -----------|*sin|-- + 2*x|
\    sin(4*x)      sin(8*x) /    \2       /
$$\frac{\sin{\left(2 x \right)}}{\left(- \frac{2 \sin^{2}{\left(2 x \right)}}{\sin{\left(4 x \right)}} + \frac{2 \sin^{2}{\left(4 x \right)}}{\sin{\left(8 x \right)}}\right) \sin{\left(2 x + \frac{\pi}{2} \right)}}$$ 
            sin(2*x)           
-------------------------------
(-tan(2*x) + tan(4*x))*cos(2*x)
$$\frac{\sin{\left(2 x \right)}}{\left(- \tan{\left(2 x \right)} + \tan{\left(4 x \right)}\right) \cos{\left(2 x \right)}}$$ 
                         2    /       2     \                     
                    4*cot (x)*\1 + cot (2*x)/                     
------------------------------------------------------------------
             2 /                  2      /       2     \\         
/       2   \  |     1       4*cot (2*x)*\1 + cot (4*x)/|         
\1 + cot (x)/ *|- -------- + ---------------------------|*cot(2*x)
               |  cot(2*x)                   2          |         
               |              /       2     \           |         
               \              \1 + cot (2*x)/ *cot(4*x) /
$$\frac{4 \left(\cot^{2}{\left(2 x \right)} + 1\right) \cot^{2}{\left(x \right)}}{\left(\cot^{2}{\left(x \right)} + 1\right)^{2} \left(- \frac{1}{\cot{\left(2 x \right)}} + \frac{4 \left(\cot^{2}{\left(4 x \right)} + 1\right) \cot^{2}{\left(2 x \right)}}{\left(\cot^{2}{\left(2 x \right)} + 1\right)^{2} \cot{\left(4 x \right)}}\right) \cot{\left(2 x \right)}}$$ 
                 2*csc(4*x)                 
--------------------------------------------
/                 2              \    2     
\-tan(2*x) + 2*sin (4*x)*csc(8*x)/*csc (2*x)
$$\frac{2 \csc{\left(4 x \right)}}{\left(2 \sin^{2}{\left(4 x \right)} \csc{\left(8 x \right)} - \tan{\left(2 x \right)}\right) \csc^{2}{\left(2 x \right)}}$$ 
                /pi      \              
             csc|-- - 2*x|              
                \2       /              
----------------------------------------
/   /pi      \      /pi      \\         
|csc|-- - 4*x|   csc|-- - 2*x||         
|   \2       /      \2       /|         
|------------- - -------------|*csc(2*x)
\   csc(4*x)        csc(2*x)  /
$$\frac{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}{\left(\frac{\csc{\left(- 4 x + \frac{\pi}{2} \right)}}{\csc{\left(4 x \right)}} - \frac{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}{\csc{\left(2 x \right)}}\right) \csc{\left(2 x \right)}}$$ 
               2*cot(x)              
-------------------------------------
/        2   \                       
\-1 + cot (x)/*(-tan(2*x) + tan(4*x))
$$\frac{2 \cot{\left(x \right)}}{\left(- \tan{\left(2 x \right)} + \tan{\left(4 x \right)}\right) \left(\cot^{2}{\left(x \right)} - 1\right)}$$ 
              1               
------------------------------
/   1          1    \         
|-------- - --------|*cot(2*x)
\cot(4*x)   cot(2*x)/
$$\frac{1}{\left(\frac{1}{\cot{\left(4 x \right)}} - \frac{1}{\cot{\left(2 x \right)}}\right) \cot{\left(2 x \right)}}$$ 
1/((1/cot(4*x) - 1/cot(2*x))*cot(2*x))
Potencias [src] 
                         /   2*I*x    -2*I*x\                       
                       I*\- e      + e      /                       
--------------------------------------------------------------------
/  /   4*I*x    -4*I*x\     /   2*I*x    -2*I*x\\                   
|I*\- e      + e      /   I*\- e      + e      /| / -2*I*x    2*I*x\
|---------------------- - ----------------------|*\e       + e     /
|    -4*I*x    4*I*x          -2*I*x    2*I*x   |                   
\   e       + e              e       + e        /
$$\frac{i \left(- e^{2 i x} + e^{- 2 i x}\right)}{\left(- \frac{i \left(- e^{2 i x} + e^{- 2 i x}\right)}{e^{2 i x} + e^{- 2 i x}} + \frac{i \left(- e^{4 i x} + e^{- 4 i x}\right)}{e^{4 i x} + e^{- 4 i x}}\right) \left(e^{2 i x} + e^{- 2 i x}\right)}$$ 
i*(-exp(2*i*x) + exp(-2*i*x))/((i*(-exp(4*i*x) + exp(-4*i*x))/(exp(-4*i*x) + exp(4*i*x)) - i*(-exp(2*i*x) + exp(-2*i*x))/(exp(-2*i*x) + exp(2*i*x)))*(exp(-2*i*x) + exp(2*i*x)))
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