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

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

Expresión a simplificar:

Solución

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