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

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

Expresión a simplificar:

Solución

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