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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
   /  p  \      /    p\
tan|-----|   tan|a - -|
   \4 + a/      \    4/
---------- + ----------
 tan(2*a)     tan(2*a)
$$\frac{\tan{\left(\frac{p}{a + 4} \right)}}{\tan{\left(2 a \right)}} + \frac{\tan{\left(a - \frac{p}{4} \right)}}{\tan{\left(2 a \right)}}$$ 
tan(p/(4 + a))/tan(2*a) + tan(a - p/4)/tan(2*a)
Simplificación general [src] 
   /  p  \      /    p\
tan|-----| + tan|a - -|
   \4 + a/      \    4/
-----------------------
        tan(2*a)
$$\frac{\tan{\left(\frac{p}{a + 4} \right)} + \tan{\left(a - \frac{p}{4} \right)}}{\tan{\left(2 a \right)}}$$ 
(tan(p/(4 + a)) + tan(a - p/4))/tan(2*a)
Respuesta numérica [src] 
tan(p/(4 + a))/tan(2*a) + tan(a - p/4)/tan(2*a)
tan(p/(4 + a))/tan(2*a) + tan(a - p/4)/tan(2*a)
Combinatoria [src] 
   /  p  \      /    p\
tan|-----| + tan|a - -|
   \4 + a/      \    4/
-----------------------
        tan(2*a)
$$\frac{\tan{\left(\frac{p}{a + 4} \right)} + \tan{\left(a - \frac{p}{4} \right)}}{\tan{\left(2 a \right)}}$$ 
(tan(p/(4 + a)) + tan(a - p/4))/tan(2*a)
Potencias [src] 
/     /    p\      /     p\\                      /    I*p     -I*p \                   
|   I*|a - -|    I*|-a + -||                      |   -----    -----|                   
|     \    4/      \     4/| / -2*I*a    2*I*a\   |   4 + a    4 + a| / -2*I*a    2*I*a\
\- e          + e          /*\e       + e     /   \- e      + e     /*\e       + e     /
----------------------------------------------- + --------------------------------------
                     /   /    p\      /     p\\                        /  I*p     -I*p \
                     | I*|a - -|    I*|-a + -||                        | -----    -----|
/   2*I*a    -2*I*a\ |   \    4/      \     4/|   /   2*I*a    -2*I*a\ | 4 + a    4 + a|
\- e      + e      /*\e          + e          /   \- e      + e      /*\e      + e     /
$$\frac{\left(e^{2 i a} + e^{- 2 i a}\right) \left(e^{i \left(- a + \frac{p}{4}\right)} - e^{i \left(a - \frac{p}{4}\right)}\right)}{\left(- e^{2 i a} + e^{- 2 i a}\right) \left(e^{i \left(- a + \frac{p}{4}\right)} + e^{i \left(a - \frac{p}{4}\right)}\right)} + \frac{\left(e^{2 i a} + e^{- 2 i a}\right) \left(- e^{\frac{i p}{a + 4}} + e^{- \frac{i p}{a + 4}}\right)}{\left(- e^{2 i a} + e^{- 2 i a}\right) \left(e^{\frac{i p}{a + 4}} + e^{- \frac{i p}{a + 4}}\right)}$$ 
(-exp(i*(a - p/4)) + exp(i*(-a + p/4)))*(exp(-2*i*a) + exp(2*i*a))/((-exp(2*i*a) + exp(-2*i*a))*(exp(i*(a - p/4)) + exp(i*(-a + p/4)))) + (-exp(i*p/(4 + a)) + exp(-i*p/(4 + a)))*(exp(-2*i*a) + exp(2*i*a))/((-exp(2*i*a) + exp(-2*i*a))*(exp(i*p/(4 + a)) + exp(-i*p/(4 + a))))
Denominador común [src] 
   /  p  \      /    p\
tan|-----| + tan|a - -|
   \4 + a/      \    4/
-----------------------
        tan(2*a)
$$\frac{\tan{\left(\frac{p}{a + 4} \right)} + \tan{\left(a - \frac{p}{4} \right)}}{\tan{\left(2 a \right)}}$$ 
(tan(p/(4 + a)) + tan(a - p/4))/tan(2*a)
Abrimos la expresión [src] 
                                 /  p  \                                               /p\                       /  p  \             2       /p\      
                              tan|-----|                3                           tan|-|             tan(a)*tan|-----|          tan (a)*tan|-|      
           tan(a)                \4 + a/             tan (a)                           \4/                       \4 + a/                     \4/      
--------------------------- + ---------- - --------------------------- - --------------------------- - ----------------- + ---------------------------
                2       /p\    2*tan(a)                    2       /p\                   2       /p\           2                           2       /p\
2*tan(a) + 2*tan (a)*tan|-|                2*tan(a) + 2*tan (a)*tan|-|   2*tan(a) + 2*tan (a)*tan|-|                       2*tan(a) + 2*tan (a)*tan|-|
                        \4/                                        \4/                           \4/                                               \4/
$$- \frac{\tan{\left(a \right)} \tan{\left(\frac{p}{a + 4} \right)}}{2} + \frac{\tan{\left(\frac{p}{a + 4} \right)}}{2 \tan{\left(a \right)}} - \frac{\tan^{3}{\left(a \right)}}{2 \tan^{2}{\left(a \right)} \tan{\left(\frac{p}{4} \right)} + 2 \tan{\left(a \right)}} + \frac{\tan^{2}{\left(a \right)} \tan{\left(\frac{p}{4} \right)}}{2 \tan^{2}{\left(a \right)} \tan{\left(\frac{p}{4} \right)} + 2 \tan{\left(a \right)}} + \frac{\tan{\left(a \right)}}{2 \tan^{2}{\left(a \right)} \tan{\left(\frac{p}{4} \right)} + 2 \tan{\left(a \right)}} - \frac{\tan{\left(\frac{p}{4} \right)}}{2 \tan^{2}{\left(a \right)} \tan{\left(\frac{p}{4} \right)} + 2 \tan{\left(a \right)}}$$ 
tan(a)/(2*tan(a) + 2*tan(a)^2*tan(p/4)) + tan(p/(4 + a))/(2*tan(a)) - tan(a)^3/(2*tan(a) + 2*tan(a)^2*tan(p/4)) - tan(p/4)/(2*tan(a) + 2*tan(a)^2*tan(p/4)) - tan(a)*tan(p/(4 + a))/2 + tan(a)^2*tan(p/4)/(2*tan(a) + 2*tan(a)^2*tan(p/4))
Denominador racional [src] 
   /  p  \      /    p\
tan|-----| + tan|a - -|
   \4 + a/      \    4/
-----------------------
        tan(2*a)
$$\frac{\tan{\left(\frac{p}{a + 4} \right)} + \tan{\left(a - \frac{p}{4} \right)}}{\tan{\left(2 a \right)}}$$ 
(tan(p/(4 + a)) + tan(a - p/4))/tan(2*a)
Unión de expresiones racionales [src] 
   /-p + 4*a\      /  p  \
tan|--------| + tan|-----|
   \   4    /      \4 + a/
--------------------------
         tan(2*a)
$$\frac{\tan{\left(\frac{p}{a + 4} \right)} + \tan{\left(\frac{4 a - p}{4} \right)}}{\tan{\left(2 a \right)}}$$ 
(tan((-p + 4*a)/4) + tan(p/(4 + a)))/tan(2*a)
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