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

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

Expresión a simplificar:

Solución

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