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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
$$\frac{\tan{\left(2 a \right)}}{- a \tan{\left(2 \right)} + \tan{\left(4 a \right)}}$$ 
tan(2*a)/(tan(4*a) - tan(2)*a)
Simplificación general [src] 
     -tan(2*a)      
--------------------
-tan(4*a) + a*tan(2)
$$- \frac{\tan{\left(2 a \right)}}{a \tan{\left(2 \right)} - \tan{\left(4 a \right)}}$$ 
-tan(2*a)/(-tan(4*a) + a*tan(2))
Respuesta numérica [src] 
tan(2*a)/(2.18503986326152*a + tan(4*a))
tan(2*a)/(2.18503986326152*a + tan(4*a))
Denominador común [src] 
     -tan(2*a)      
--------------------
-tan(4*a) + a*tan(2)
$$- \frac{\tan{\left(2 a \right)}}{a \tan{\left(2 \right)} - \tan{\left(4 a \right)}}$$ 
-tan(2*a)/(-tan(4*a) + a*tan(2))
Potencias [src] 
                        /   2*I*a    -2*I*a\                      
                      I*\- e      + e      /                      
------------------------------------------------------------------
/  /   4*I*a    -4*I*a\       /   2*I    -2*I\\                   
|I*\- e      + e      /   I*a*\- e    + e    /| / -2*I*a    2*I*a\
|---------------------- - --------------------|*\e       + e     /
|    -4*I*a    4*I*a           -2*I    2*I    |                   
\   e       + e               e     + e       /
$$\frac{i \left(- e^{2 i a} + e^{- 2 i a}\right)}{\left(- \frac{i a \left(e^{- 2 i} - e^{2 i}\right)}{e^{- 2 i} + e^{2 i}} + \frac{i \left(- e^{4 i a} + e^{- 4 i a}\right)}{e^{4 i a} + e^{- 4 i a}}\right) \left(e^{2 i a} + e^{- 2 i a}\right)}$$ 
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*a*(-exp(2*i) + exp(-2*i))/(exp(-2*i) + exp(2*i)))*(exp(-2*i*a) + exp(2*i*a)))
Combinatoria [src] 
     -tan(2*a)      
--------------------
-tan(4*a) + a*tan(2)
$$- \frac{\tan{\left(2 a \right)}}{a \tan{\left(2 \right)} - \tan{\left(4 a \right)}}$$ 
-tan(2*a)/(-tan(4*a) + a*tan(2))
Abrimos la expresión [src] 
                                                    2*tan(a)                                                    
----------------------------------------------------------------------------------------------------------------
              3                         5                                                            2          
         8*tan (a)                 4*tan (a)                  4*tan(a)           2*a*tan(1)   2*a*tan (a)*tan(1)
- ----------------------- + ----------------------- + ----------------------- - ----------- + ------------------
         4           2             4           2             4           2             2                2       
  1 + tan (a) - 6*tan (a)   1 + tan (a) - 6*tan (a)   1 + tan (a) - 6*tan (a)   1 - tan (1)      1 - tan (1)
$$\frac{2 \tan{\left(a \right)}}{\frac{2 a \tan{\left(1 \right)} \tan^{2}{\left(a \right)}}{1 - \tan^{2}{\left(1 \right)}} - \frac{2 a \tan{\left(1 \right)}}{1 - \tan^{2}{\left(1 \right)}} + \frac{4 \tan^{5}{\left(a \right)}}{\tan^{4}{\left(a \right)} - 6 \tan^{2}{\left(a \right)} + 1} - \frac{8 \tan^{3}{\left(a \right)}}{\tan^{4}{\left(a \right)} - 6 \tan^{2}{\left(a \right)} + 1} + \frac{4 \tan{\left(a \right)}}{\tan^{4}{\left(a \right)} - 6 \tan^{2}{\left(a \right)} + 1}}$$ 
2*tan(a)/(-8*tan(a)^3/(1 + tan(a)^4 - 6*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) - 2*a*tan(1)/(1 - tan(1)^2) + 2*a*tan(a)^2*tan(1)/(1 - tan(1)^2))
Parte trigonométrica [src] 
                                 2    /       2     \                            
                            4*tan (a)*\1 + tan (2*a)/                            
---------------------------------------------------------------------------------
             2 /     2      /       2     \          2    /       2   \\         
/       2   \  |4*tan (2*a)*\1 + tan (4*a)/   4*a*tan (1)*\1 + tan (2)/|         
\1 + tan (a)/ *|--------------------------- - -------------------------|*tan(2*a)
               |                2                            2         |         
               | /       2     \                /       2   \          |         
               \ \1 + tan (2*a)/ *tan(4*a)      \1 + tan (1)/ *tan(2)  /
$$\frac{4 \left(\tan^{2}{\left(2 a \right)} + 1\right) \tan^{2}{\left(a \right)}}{\left(- \frac{4 a \left(1 + \tan^{2}{\left(2 \right)}\right) \tan^{2}{\left(1 \right)}}{\left(1 + \tan^{2}{\left(1 \right)}\right)^{2} \tan{\left(2 \right)}} + \frac{4 \left(\tan^{2}{\left(4 a \right)} + 1\right) \tan^{2}{\left(2 a \right)}}{\left(\tan^{2}{\left(2 a \right)} + 1\right)^{2} \tan{\left(4 a \right)}}\right) \left(\tan^{2}{\left(a \right)} + 1\right)^{2} \tan{\left(2 a \right)}}$$ 
           sec(2*a)           
------------------------------
/sec(4*a)   a*sec(2)\         
|-------- - --------|*csc(2*a)
\csc(4*a)    csc(2) /
$$\frac{\sec{\left(2 a \right)}}{\left(- \frac{a \sec{\left(2 \right)}}{\csc{\left(2 \right)}} + \frac{\sec{\left(4 a \right)}}{\csc{\left(4 a \right)}}\right) \csc{\left(2 a \right)}}$$ 
                 2                  
            2*sin (2*a)             
------------------------------------
/     2               2   \         
|2*sin (4*a)   2*a*sin (2)|         
|----------- - -----------|*sin(4*a)
\  sin(8*a)       sin(4)  /
$$\frac{2 \sin^{2}{\left(2 a \right)}}{\left(- \frac{2 a \sin^{2}{\left(2 \right)}}{\sin{\left(4 \right)}} + \frac{2 \sin^{2}{\left(4 a \right)}}{\sin{\left(8 a \right)}}\right) \sin{\left(4 a \right)}}$$ 
             2*csc(4*a)            
-----------------------------------
/2*csc(8*a)   2*a*csc(4)\    2     
|---------- - ----------|*csc (2*a)
|   2             2     |          
\csc (4*a)     csc (2)  /
$$\frac{2 \csc{\left(4 a \right)}}{\left(- \frac{2 a \csc{\left(4 \right)}}{\csc^{2}{\left(2 \right)}} + \frac{2 \csc{\left(8 a \right)}}{\csc^{2}{\left(4 a \right)}}\right) \csc^{2}{\left(2 a \right)}}$$ 
                  2                   
             2*sin (2*a)              
--------------------------------------
/      2                2   \         
| 2*sin (4*a)    2*a*sin (2)|         
|------------- - -----------|*sin(4*a)
|   /      pi\      sin(4)  |         
|cos|8*a - --|              |         
\   \      2 /              /
$$\frac{2 \sin^{2}{\left(2 a \right)}}{\left(- \frac{2 a \sin^{2}{\left(2 \right)}}{\sin{\left(4 \right)}} + \frac{2 \sin^{2}{\left(4 a \right)}}{\cos{\left(8 a - \frac{\pi}{2} \right)}}\right) \sin{\left(4 a \right)}}$$ 
                      2*csc(4*a)                     
-----------------------------------------------------
/     2                        2          \    2     
\2*sin (4*a)*csc(8*a) - 2*a*sin (2)*csc(4)/*csc (2*a)
$$\frac{2 \csc{\left(4 a \right)}}{\left(- 2 a \sin^{2}{\left(2 \right)} \csc{\left(4 \right)} + 2 \sin^{2}{\left(4 a \right)} \csc{\left(8 a \right)}\right) \csc^{2}{\left(2 a \right)}}$$ 
                                 2    /       2     \                            
                            4*cot (a)*\1 + cot (2*a)/                            
---------------------------------------------------------------------------------
             2 /     2      /       2     \          2    /       2   \\         
/       2   \  |4*cot (2*a)*\1 + cot (4*a)/   4*a*cot (1)*\1 + cot (2)/|         
\1 + cot (a)/ *|--------------------------- - -------------------------|*cot(2*a)
               |                2                            2         |         
               | /       2     \                /       2   \          |         
               \ \1 + cot (2*a)/ *cot(4*a)      \1 + cot (1)/ *cot(2)  /
$$\frac{4 \left(\cot^{2}{\left(2 a \right)} + 1\right) \cot^{2}{\left(a \right)}}{\left(- \frac{4 a \left(\cot^{2}{\left(2 \right)} + 1\right) \cot^{2}{\left(1 \right)}}{\left(\cot^{2}{\left(1 \right)} + 1\right)^{2} \cot{\left(2 \right)}} + \frac{4 \left(\cot^{2}{\left(4 a \right)} + 1\right) \cot^{2}{\left(2 a \right)}}{\left(\cot^{2}{\left(2 a \right)} + 1\right)^{2} \cot{\left(4 a \right)}}\right) \left(\cot^{2}{\left(a \right)} + 1\right)^{2} \cot{\left(2 a \right)}}$$ 
                 /      pi\               
             -cos|2*a - --|               
                 \      2 /               
------------------------------------------
/     /      pi\        /    pi\\         
|  cos|4*a - --|   a*cos|2 - --||         
|     \      2 /        \    2 /|         
|- ------------- + -------------|*cos(2*a)
\     cos(4*a)         cos(2)   /
$$- \frac{\cos{\left(2 a - \frac{\pi}{2} \right)}}{\left(\frac{a \cos{\left(2 - \frac{\pi}{2} \right)}}{\cos{\left(2 \right)}} - \frac{\cos{\left(4 a - \frac{\pi}{2} \right)}}{\cos{\left(4 a \right)}}\right) \cos{\left(2 a \right)}}$$ 
                        2                         
                   2*sin (2*a)                    
--------------------------------------------------
/                                   2   \         
|     2         /      pi\   2*a*sin (2)|         
|2*sin (4*a)*sec|8*a - --| - -----------|*sin(4*a)
\               \      2 /      sin(4)  /
$$\frac{2 \sin^{2}{\left(2 a \right)}}{\left(- \frac{2 a \sin^{2}{\left(2 \right)}}{\sin{\left(4 \right)}} + 2 \sin^{2}{\left(4 a \right)} \sec{\left(8 a - \frac{\pi}{2} \right)}\right) \sin{\left(4 a \right)}}$$ 
             1              
----------------------------
/   1         a   \         
|-------- - ------|*cot(2*a)
\cot(4*a)   cot(2)/
$$\frac{1}{\left(- \frac{a}{\cot{\left(2 \right)}} + \frac{1}{\cot{\left(4 a \right)}}\right) \cot{\left(2 a \right)}}$$ 
                  -sec(2*a)                  
---------------------------------------------
/     sec(4*a)       a*sec(2) \    /      pi\
|- ------------- + -----------|*sec|2*a - --|
|     /      pi\      /    pi\|    \      2 /
|  sec|4*a - --|   sec|2 - --||              
\     \      2 /      \    2 //
$$- \frac{\sec{\left(2 a \right)}}{\left(\frac{a \sec{\left(2 \right)}}{\sec{\left(2 - \frac{\pi}{2} \right)}} - \frac{\sec{\left(4 a \right)}}{\sec{\left(4 a - \frac{\pi}{2} \right)}}\right) \sec{\left(2 a - \frac{\pi}{2} \right)}}$$ 
                       2/      pi\                 
                  2*cos |2*a - --|                 
                        \      2 /                 
---------------------------------------------------
/     2/      pi\          2/    pi\\              
|2*cos |4*a - --|   2*a*cos |2 - --||              
|      \      2 /           \    2 /|    /      pi\
|---------------- - ----------------|*cos|4*a - --|
|    /      pi\          /    pi\   |    \      2 /
| cos|8*a - --|       cos|4 - --|   |              
\    \      2 /          \    2 /   /
$$\frac{2 \cos^{2}{\left(2 a - \frac{\pi}{2} \right)}}{\left(- \frac{2 a \cos^{2}{\left(2 - \frac{\pi}{2} \right)}}{\cos{\left(4 - \frac{\pi}{2} \right)}} + \frac{2 \cos^{2}{\left(4 a - \frac{\pi}{2} \right)}}{\cos{\left(8 a - \frac{\pi}{2} \right)}}\right) \cos{\left(4 a - \frac{\pi}{2} \right)}}$$ 
                  sec(2*a)                 
-------------------------------------------
/   sec(4*a)       a*sec(2) \    /      pi\
|------------- - -----------|*sec|2*a - --|
|   /      pi\      /    pi\|    \      2 /
|sec|4*a - --|   sec|2 - --||              
\   \      2 /      \    2 //
$$\frac{\sec{\left(2 a \right)}}{\left(- \frac{a \sec{\left(2 \right)}}{\sec{\left(2 - \frac{\pi}{2} \right)}} + \frac{\sec{\left(4 a \right)}}{\sec{\left(4 a - \frac{\pi}{2} \right)}}\right) \sec{\left(2 a - \frac{\pi}{2} \right)}}$$ 
           sin(2*a)           
------------------------------
/sin(4*a)   a*sin(2)\         
|-------- - --------|*cos(2*a)
\cos(4*a)    cos(2) /
$$\frac{\sin{\left(2 a \right)}}{\left(- \frac{a \sin{\left(2 \right)}}{\cos{\left(2 \right)}} + \frac{\sin{\left(4 a \right)}}{\cos{\left(4 a \right)}}\right) \cos{\left(2 a \right)}}$$ 
                 /pi      \              
              csc|-- - 2*a|              
                 \2       /              
-----------------------------------------
/   /pi      \        /     pi\\         
|csc|-- - 4*a|   a*csc|-2 + --||         
|   \2       /        \     2 /|         
|------------- - --------------|*csc(2*a)
\   csc(4*a)         csc(2)    /
$$\frac{\csc{\left(- 2 a + \frac{\pi}{2} \right)}}{\left(- \frac{a \csc{\left(-2 + \frac{\pi}{2} \right)}}{\csc{\left(2 \right)}} + \frac{\csc{\left(- 4 a + \frac{\pi}{2} \right)}}{\csc{\left(4 a \right)}}\right) \csc{\left(2 a \right)}}$$ 
                /      pi\              
             cos|2*a - --|              
                \      2 /              
----------------------------------------
/   /      pi\        /    pi\\         
|cos|4*a - --|   a*cos|2 - --||         
|   \      2 /        \    2 /|         
|------------- - -------------|*cos(2*a)
\   cos(4*a)         cos(2)   /
$$\frac{\cos{\left(2 a - \frac{\pi}{2} \right)}}{\left(- \frac{a \cos{\left(2 - \frac{\pi}{2} \right)}}{\cos{\left(2 \right)}} + \frac{\cos{\left(4 a - \frac{\pi}{2} \right)}}{\cos{\left(4 a \right)}}\right) \cos{\left(2 a \right)}}$$ 
     -tan(2*a)      
--------------------
-tan(4*a) + a*tan(2)
$$- \frac{\tan{\left(2 a \right)}}{a \tan{\left(2 \right)} - \tan{\left(4 a \right)}}$$ 
                      /      pi\                  
                 2*sec|4*a - --|                  
                      \      2 /                  
--------------------------------------------------
/     /      pi\          /    pi\\               
|2*sec|8*a - --|   2*a*sec|4 - --||               
|     \      2 /          \    2 /|    2/      pi\
|--------------- - ---------------|*sec |2*a - --|
|    2/      pi\        2/    pi\ |     \      2 /
| sec |4*a - --|     sec |2 - --| |               
\     \      2 /         \    2 / /
$$\frac{2 \sec{\left(4 a - \frac{\pi}{2} \right)}}{\left(- \frac{2 a \sec{\left(4 - \frac{\pi}{2} \right)}}{\sec^{2}{\left(2 - \frac{\pi}{2} \right)}} + \frac{2 \sec{\left(8 a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(4 a - \frac{\pi}{2} \right)}}\right) \sec^{2}{\left(2 a - \frac{\pi}{2} \right)}}$$ 
                     2       4                            
                4*cot (a)*sin (a)*tan(2*a)                
----------------------------------------------------------
/              sin(4*a)                        \    2     
|----------------------------------- - a*tan(2)|*sin (2*a)
|                     2                        |          
\-3 + 2*(1 - cos(2*a))  + 4*cos(2*a)           /
$$\frac{4 \sin^{4}{\left(a \right)} \tan{\left(2 a \right)} \cot^{2}{\left(a \right)}}{\left(- a \tan{\left(2 \right)} + \frac{\sin{\left(4 a \right)}}{2 \left(1 - \cos{\left(2 a \right)}\right)^{2} + 4 \cos{\left(2 a \right)} - 3}\right) \sin^{2}{\left(2 a \right)}}$$ 
                   2                  
             -2*sin (2*a)             
--------------------------------------
/       2               2   \         
|  2*sin (4*a)   2*a*sin (2)|         
|- ----------- + -----------|*sin(4*a)
\    sin(8*a)       sin(4)  /
$$- \frac{2 \sin^{2}{\left(2 a \right)}}{\left(\frac{2 a \sin^{2}{\left(2 \right)}}{\sin{\left(4 \right)}} - \frac{2 \sin^{2}{\left(4 a \right)}}{\sin{\left(8 a \right)}}\right) \sin{\left(4 a \right)}}$$ 
                  /pi      \               
              -csc|-- - 2*a|               
                  \2       /               
-------------------------------------------
/     /pi      \        /     pi\\         
|  csc|-- - 4*a|   a*csc|-2 + --||         
|     \2       /        \     2 /|         
|- ------------- + --------------|*csc(2*a)
\     csc(4*a)         csc(2)    /
$$- \frac{\csc{\left(- 2 a + \frac{\pi}{2} \right)}}{\left(\frac{a \csc{\left(-2 + \frac{\pi}{2} \right)}}{\csc{\left(2 \right)}} - \frac{\csc{\left(- 4 a + \frac{\pi}{2} \right)}}{\csc{\left(4 a \right)}}\right) \csc{\left(2 a \right)}}$$ 
                                    4       2                                   
                               4*cos (a)*tan (a)                                
--------------------------------------------------------------------------------
/                                                  2        \                   
|              sin(4*a)                     8*a*tan (1)     |    2              
|----------------------------------- - ---------------------|*cos (2*a)*tan(2*a)
|                     2                             2       |                   
|-3 + 2*(1 - cos(2*a))  + 4*cos(2*a)   /       2   \        |                   
\                                      \1 + tan (1)/ *sin(4)/
$$\frac{4 \cos^{4}{\left(a \right)} \tan^{2}{\left(a \right)}}{\left(- \frac{8 a \tan^{2}{\left(1 \right)}}{\left(1 + \tan^{2}{\left(1 \right)}\right)^{2} \sin{\left(4 \right)}} + \frac{\sin{\left(4 a \right)}}{2 \left(1 - \cos{\left(2 a \right)}\right)^{2} + 4 \cos{\left(2 a \right)} - 3}\right) \cos^{2}{\left(2 a \right)} \tan{\left(2 a \right)}}$$ 
             -1               
------------------------------
/     1         a   \         
|- -------- + ------|*cot(2*a)
\  cot(4*a)   cot(2)/
$$- \frac{1}{\left(\frac{a}{\cot{\left(2 \right)}} - \frac{1}{\cot{\left(4 a \right)}}\right) \cot{\left(2 a \right)}}$$ 
-1/((-1/cot(4*a) + a/cot(2))*cot(2*a))
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