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Simplificación de expresiones/ Descomposición en fracciones simples/ sin(a+b+c)/((sin(a)*sin(b)*sin(c)))-(tan(a)+tan(b)+tan(c))/(tan(a)tan(b)tan(c))

¿Cómo vas a descomponer esta sin(a+b+c)/((sin(a)*sin(b)*sin(c)))-(tan(a)+tan(b)+tan(c))/(tan(a)tan(b)tan(c)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
   sin(a + b + c)      tan(a) + tan(b) + tan(c)
-------------------- - ------------------------
sin(a)*sin(b)*sin(c)     tan(a)*tan(b)*tan(c)
−(tan⁡(a)+tan⁡(b))+tan⁡(c)tan⁡(a)tan⁡(b)tan⁡(c)+sin⁡(c+(a+b))sin⁡(a)sin⁡(b)sin⁡(c)- \frac{\left(\tan{\left(a \right)} + \tan{\left(b \right)}\right) + \tan{\left(c \right)}}{\tan{\left(a \right)} \tan{\left(b \right)} \tan{\left(c \right)}} + \frac{\sin{\left(c + \left(a + b\right) \right)}}{\sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)}} 
sin(a + b + c)/(((sin(a)*sin(b))*sin(c))) - (tan(a) + tan(b) + tan(c))/((tan(a)*tan(b))*tan(c))
Simplificación general [src] 
-1
−1-1 
-1
Respuesta numérica [src] 
sin(a + b + c)/(sin(a)*sin(b)*sin(c)) - (tan(a) + tan(b) + tan(c))/(tan(a)*tan(b)*tan(c))
sin(a + b + c)/(sin(a)*sin(b)*sin(c)) - (tan(a) + tan(b) + tan(c))/(tan(a)*tan(b)*tan(c))
Denominador racional [src] 
sin(a + b + c)*tan(a)*tan(b)*tan(c) - sin(a)*sin(b)*sin(c)*tan(a) - sin(a)*sin(b)*sin(c)*tan(b) - sin(a)*sin(b)*sin(c)*tan(c)
-----------------------------------------------------------------------------------------------------------------------------
                                          sin(a)*sin(b)*sin(c)*tan(a)*tan(b)*tan(c)
−sin⁡(a)sin⁡(b)sin⁡(c)tan⁡(a)−sin⁡(a)sin⁡(b)sin⁡(c)tan⁡(b)−sin⁡(a)sin⁡(b)sin⁡(c)tan⁡(c)+sin⁡(a+b+c)tan⁡(a)tan⁡(b)tan⁡(c)sin⁡(a)sin⁡(b)sin⁡(c)tan⁡(a)tan⁡(b)tan⁡(c)\frac{- \sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} \tan{\left(a \right)} - \sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} \tan{\left(b \right)} - \sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} \tan{\left(c \right)} + \sin{\left(a + b + c \right)} \tan{\left(a \right)} \tan{\left(b \right)} \tan{\left(c \right)}}{\sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} \tan{\left(a \right)} \tan{\left(b \right)} \tan{\left(c \right)}} 
(sin(a + b + c)*tan(a)*tan(b)*tan(c) - sin(a)*sin(b)*sin(c)*tan(a) - sin(a)*sin(b)*sin(c)*tan(b) - sin(a)*sin(b)*sin(c)*tan(c))/(sin(a)*sin(b)*sin(c)*tan(a)*tan(b)*tan(c))
Potencias [src] 
   sin(a + b + c)      tan(a) + tan(b) + tan(c)
-------------------- - ------------------------
sin(a)*sin(b)*sin(c)     tan(a)*tan(b)*tan(c)
−tan⁡(a)+tan⁡(b)+tan⁡(c)tan⁡(a)tan⁡(b)tan⁡(c)+sin⁡(a+b+c)sin⁡(a)sin⁡(b)sin⁡(c)- \frac{\tan{\left(a \right)} + \tan{\left(b \right)} + \tan{\left(c \right)}}{\tan{\left(a \right)} \tan{\left(b \right)} \tan{\left(c \right)}} + \frac{\sin{\left(a + b + c \right)}}{\sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)}} 
   sin(a + b + c)      -tan(a) - tan(b) - tan(c)
-------------------- + -------------------------
sin(a)*sin(b)*sin(c)      tan(a)*tan(b)*tan(c)
−tan⁡(a)−tan⁡(b)−tan⁡(c)tan⁡(a)tan⁡(b)tan⁡(c)+sin⁡(a+b+c)sin⁡(a)sin⁡(b)sin⁡(c)\frac{- \tan{\left(a \right)} - \tan{\left(b \right)} - \tan{\left(c \right)}}{\tan{\left(a \right)} \tan{\left(b \right)} \tan{\left(c \right)}} + \frac{\sin{\left(a + b + c \right)}}{\sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)}} 
                                                                                                      /  /   I*a    -I*a\     /   I*b    -I*b\     /   I*c    -I*c\\
                                                         / I*a    -I*a\ / I*b    -I*b\ / I*c    -I*c\ |I*\- e    + e    /   I*\- e    + e    /   I*\- e    + e    /|
                                                       I*\e    + e    /*\e    + e    /*\e    + e    /*|------------------ + ------------------ + ------------------|
          /   I*(-a - b - c)    I*(a + b + c)\                                                        |    I*a    -I*a          I*b    -I*b          I*c    -I*c   |
        4*\- e               + e             /                                                        \   e    + e             e    + e             e    + e       /
- -------------------------------------------------- - -------------------------------------------------------------------------------------------------------------
  /   -I*a    I*a\ /   -I*b    I*b\ /   -I*c    I*c\                                 /   I*a    -I*a\ /   I*b    -I*b\ /   I*c    -I*c\                             
  \- e     + e   /*\- e     + e   /*\- e     + e   /                                 \- e    + e    /*\- e    + e    /*\- e    + e    /
−4(−ei(−a−b−c)+ei(a+b+c))(eia−e−ia)(eib−e−ib)(eic−e−ic)−i(eia+e−ia)(eib+e−ib)(eic+e−ic)(i(−eia+e−ia)eia+e−ia+i(−eib+e−ib)eib+e−ib+i(−eic+e−ic)eic+e−ic)(−eia+e−ia)(−eib+e−ib)(−eic+e−ic)- \frac{4 \left(- e^{i \left(- a - b - c\right)} + e^{i \left(a + b + c\right)}\right)}{\left(e^{i a} - e^{- i a}\right) \left(e^{i b} - e^{- i b}\right) \left(e^{i c} - e^{- i c}\right)} - \frac{i \left(e^{i a} + e^{- i a}\right) \left(e^{i b} + e^{- i b}\right) \left(e^{i c} + e^{- i c}\right) \left(\frac{i \left(- e^{i a} + e^{- i a}\right)}{e^{i a} + e^{- i a}} + \frac{i \left(- e^{i b} + e^{- i b}\right)}{e^{i b} + e^{- i b}} + \frac{i \left(- e^{i c} + e^{- i c}\right)}{e^{i c} + e^{- i c}}\right)}{\left(- e^{i a} + e^{- i a}\right) \left(- e^{i b} + e^{- i b}\right) \left(- e^{i c} + e^{- i c}\right)} 
-4*(-exp(i*(-a - b - c)) + exp(i*(a + b + c)))/((-exp(-i*a) + exp(i*a))*(-exp(-i*b) + exp(i*b))*(-exp(-i*c) + exp(i*c))) - i*(exp(i*a) + exp(-i*a))*(exp(i*b) + exp(-i*b))*(exp(i*c) + exp(-i*c))*(i*(-exp(i*a) + exp(-i*a))/(exp(i*a) + exp(-i*a)) + i*(-exp(i*b) + exp(-i*b))/(exp(i*b) + exp(-i*b)) + i*(-exp(i*c) + exp(-i*c))/(exp(i*c) + exp(-i*c)))/((-exp(i*a) + exp(-i*a))*(-exp(i*b) + exp(-i*b))*(-exp(i*c) + exp(-i*c)))
Denominador común [src] 
sin(a + b + c)*tan(a)*tan(b)*tan(c) - sin(a)*sin(b)*sin(c)*tan(a) - sin(a)*sin(b)*sin(c)*tan(b) - sin(a)*sin(b)*sin(c)*tan(c)
-----------------------------------------------------------------------------------------------------------------------------
                                          sin(a)*sin(b)*sin(c)*tan(a)*tan(b)*tan(c)
−sin⁡(a)sin⁡(b)sin⁡(c)tan⁡(a)−sin⁡(a)sin⁡(b)sin⁡(c)tan⁡(b)−sin⁡(a)sin⁡(b)sin⁡(c)tan⁡(c)+sin⁡(a+b+c)tan⁡(a)tan⁡(b)tan⁡(c)sin⁡(a)sin⁡(b)sin⁡(c)tan⁡(a)tan⁡(b)tan⁡(c)\frac{- \sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} \tan{\left(a \right)} - \sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} \tan{\left(b \right)} - \sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} \tan{\left(c \right)} + \sin{\left(a + b + c \right)} \tan{\left(a \right)} \tan{\left(b \right)} \tan{\left(c \right)}}{\sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} \tan{\left(a \right)} \tan{\left(b \right)} \tan{\left(c \right)}} 
(sin(a + b + c)*tan(a)*tan(b)*tan(c) - sin(a)*sin(b)*sin(c)*tan(a) - sin(a)*sin(b)*sin(c)*tan(b) - sin(a)*sin(b)*sin(c)*tan(c))/(sin(a)*sin(b)*sin(c)*tan(a)*tan(b)*tan(c))
Unión de expresiones racionales [src] 
sin(a + b + c)*tan(a)*tan(b)*tan(c) - (tan(a) + tan(b) + tan(c))*sin(a)*sin(b)*sin(c)
-------------------------------------------------------------------------------------
                      sin(a)*sin(b)*sin(c)*tan(a)*tan(b)*tan(c)
−(tan⁡(a)+tan⁡(b)+tan⁡(c))sin⁡(a)sin⁡(b)sin⁡(c)+sin⁡(a+b+c)tan⁡(a)tan⁡(b)tan⁡(c)sin⁡(a)sin⁡(b)sin⁡(c)tan⁡(a)tan⁡(b)tan⁡(c)\frac{- \left(\tan{\left(a \right)} + \tan{\left(b \right)} + \tan{\left(c \right)}\right) \sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} + \sin{\left(a + b + c \right)} \tan{\left(a \right)} \tan{\left(b \right)} \tan{\left(c \right)}}{\sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} \tan{\left(a \right)} \tan{\left(b \right)} \tan{\left(c \right)}} 
(sin(a + b + c)*tan(a)*tan(b)*tan(c) - (tan(a) + tan(b) + tan(c))*sin(a)*sin(b)*sin(c))/(sin(a)*sin(b)*sin(c)*tan(a)*tan(b)*tan(c))
Combinatoria [src] 
sin(a + b + c)*tan(a)*tan(b)*tan(c) - sin(a)*sin(b)*sin(c)*tan(a) - sin(a)*sin(b)*sin(c)*tan(b) - sin(a)*sin(b)*sin(c)*tan(c)
-----------------------------------------------------------------------------------------------------------------------------
                                          sin(a)*sin(b)*sin(c)*tan(a)*tan(b)*tan(c)
−sin⁡(a)sin⁡(b)sin⁡(c)tan⁡(a)−sin⁡(a)sin⁡(b)sin⁡(c)tan⁡(b)−sin⁡(a)sin⁡(b)sin⁡(c)tan⁡(c)+sin⁡(a+b+c)tan⁡(a)tan⁡(b)tan⁡(c)sin⁡(a)sin⁡(b)sin⁡(c)tan⁡(a)tan⁡(b)tan⁡(c)\frac{- \sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} \tan{\left(a \right)} - \sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} \tan{\left(b \right)} - \sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} \tan{\left(c \right)} + \sin{\left(a + b + c \right)} \tan{\left(a \right)} \tan{\left(b \right)} \tan{\left(c \right)}}{\sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} \tan{\left(a \right)} \tan{\left(b \right)} \tan{\left(c \right)}} 
(sin(a + b + c)*tan(a)*tan(b)*tan(c) - sin(a)*sin(b)*sin(c)*tan(a) - sin(a)*sin(b)*sin(c)*tan(b) - sin(a)*sin(b)*sin(c)*tan(c))/(sin(a)*sin(b)*sin(c)*tan(a)*tan(b)*tan(c))
Compilar la expresión [src] 
   sin(a + b + c)      tan(a) + tan(b) + tan(c)
-------------------- - ------------------------
sin(a)*sin(b)*sin(c)     tan(a)*tan(b)*tan(c)
−tan⁡(a)+tan⁡(b)+tan⁡(c)tan⁡(a)tan⁡(b)tan⁡(c)+sin⁡(c+(a+b))sin⁡(a)sin⁡(b)sin⁡(c)- \frac{\tan{\left(a \right)} + \tan{\left(b \right)} + \tan{\left(c \right)}}{\tan{\left(a \right)} \tan{\left(b \right)} \tan{\left(c \right)}} + \frac{\sin{\left(c + \left(a + b\right) \right)}}{\sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)}} 
sin(a + b + c)/(sin(a)*sin(b)*sin(c)) - (tan(a) + tan(b) + tan(c))/(tan(a)*tan(b)*tan(c))
Abrimos la expresión [src] 
           1               1               1         cos(a)*cos(b)   cos(a)*cos(c)   cos(b)*cos(c)
-1 - ------------- - ------------- - ------------- + ------------- + ------------- + -------------
     tan(a)*tan(b)   tan(a)*tan(c)   tan(b)*tan(c)   sin(a)*sin(b)   sin(a)*sin(c)   sin(b)*sin(c)
−1−1tan⁡(b)tan⁡(c)−1tan⁡(a)tan⁡(c)−1tan⁡(a)tan⁡(b)+cos⁡(b)cos⁡(c)sin⁡(b)sin⁡(c)+cos⁡(a)cos⁡(c)sin⁡(a)sin⁡(c)+cos⁡(a)cos⁡(b)sin⁡(a)sin⁡(b)-1 - \frac{1}{\tan{\left(b \right)} \tan{\left(c \right)}} - \frac{1}{\tan{\left(a \right)} \tan{\left(c \right)}} - \frac{1}{\tan{\left(a \right)} \tan{\left(b \right)}} + \frac{\cos{\left(b \right)} \cos{\left(c \right)}}{\sin{\left(b \right)} \sin{\left(c \right)}} + \frac{\cos{\left(a \right)} \cos{\left(c \right)}}{\sin{\left(a \right)} \sin{\left(c \right)}} + \frac{\cos{\left(a \right)} \cos{\left(b \right)}}{\sin{\left(a \right)} \sin{\left(b \right)}} 
-1 - 1/(tan(a)*tan(b)) - 1/(tan(a)*tan(c)) - 1/(tan(b)*tan(c)) + cos(a)*cos(b)/(sin(a)*sin(b)) + cos(a)*cos(c)/(sin(a)*sin(c)) + cos(b)*cos(c)/(sin(b)*sin(c))
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