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Simplificación de expresiones/ Descomposición en fracciones simples/ sin(a)/(1+cos(a))+(1+cot(x))/sin(a)

¿Cómo vas a descomponer esta sin(a)/(1+cos(a))+(1+cot(x))/sin(a) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
  sin(a)     1 + cot(x)
---------- + ----------
1 + cos(a)     sin(a)
$$\frac{\cot{\left(x \right)} + 1}{\sin{\left(a \right)}} + \frac{\sin{\left(a \right)}}{\cos{\left(a \right)} + 1}$$ 
sin(a)/(1 + cos(a)) + (1 + cot(x))/sin(a)
Simplificación general [src] 
   2                               
sin (a) + (1 + cos(a))*(1 + cot(x))
-----------------------------------
        (1 + cos(a))*sin(a)
$$\frac{\left(\cos{\left(a \right)} + 1\right) \left(\cot{\left(x \right)} + 1\right) + \sin^{2}{\left(a \right)}}{\left(\cos{\left(a \right)} + 1\right) \sin{\left(a \right)}}$$ 
(sin(a)^2 + (1 + cos(a))*(1 + cot(x)))/((1 + cos(a))*sin(a))
Respuesta numérica [src] 
sin(a)/(1.0 + cos(a)) + (1.0 + cot(x))/sin(a)
sin(a)/(1.0 + cos(a)) + (1.0 + cot(x))/sin(a)
Unión de expresiones racionales [src] 
   2                               
sin (a) + (1 + cos(a))*(1 + cot(x))
-----------------------------------
        (1 + cos(a))*sin(a)
$$\frac{\left(\cos{\left(a \right)} + 1\right) \left(\cot{\left(x \right)} + 1\right) + \sin^{2}{\left(a \right)}}{\left(\cos{\left(a \right)} + 1\right) \sin{\left(a \right)}}$$ 
(sin(a)^2 + (1 + cos(a))*(1 + cot(x)))/((1 + cos(a))*sin(a))
Potencias [src] 
                      /   -I*a    I*a\ 
2*I*(1 + cot(x))    I*\- e     + e   / 
---------------- - --------------------
    -I*a    I*a      /     I*a    -I*a\
 - e     + e         |    e      e    |
                   2*|1 + ---- + -----|
                     \     2       2  /
$$- \frac{i \left(e^{i a} - e^{- i a}\right)}{2 \left(\frac{e^{i a}}{2} + 1 + \frac{e^{- i a}}{2}\right)} + \frac{2 i \left(\cot{\left(x \right)} + 1\right)}{e^{i a} - e^{- i a}}$$ 
2*i*(1 + cot(x))/(-exp(-i*a) + exp(i*a)) - i*(-exp(-i*a) + exp(i*a))/(2*(1 + exp(i*a)/2 + exp(-i*a)/2))
Combinatoria [src] 
       2                                     
1 + sin (a) + cos(a)*cot(x) + cos(a) + cot(x)
---------------------------------------------
             (1 + cos(a))*sin(a)
$$\frac{\sin^{2}{\left(a \right)} + \cos{\left(a \right)} \cot{\left(x \right)} + \cos{\left(a \right)} + \cot{\left(x \right)} + 1}{\left(\cos{\left(a \right)} + 1\right) \sin{\left(a \right)}}$$ 
(1 + sin(a)^2 + cos(a)*cot(x) + cos(a) + cot(x))/((1 + cos(a))*sin(a))
Abrimos la expresión [src] 
  1      cot(x)     sin(a)  
------ + ------ + ----------
sin(a)   sin(a)   1 + cos(a)
$$\frac{\cot{\left(x \right)}}{\sin{\left(a \right)}} + \frac{1}{\sin{\left(a \right)}} + \frac{\sin{\left(a \right)}}{\cos{\left(a \right)} + 1}$$ 
1/sin(a) + cot(x)/sin(a) + sin(a)/(1 + cos(a))
Denominador racional [src] 
   2                               
sin (a) + (1 + cos(a))*(1 + cot(x))
-----------------------------------
        (1 + cos(a))*sin(a)
$$\frac{\left(\cos{\left(a \right)} + 1\right) \left(\cot{\left(x \right)} + 1\right) + \sin^{2}{\left(a \right)}}{\left(\cos{\left(a \right)} + 1\right) \sin{\left(a \right)}}$$ 
(sin(a)^2 + (1 + cos(a))*(1 + cot(x)))/((1 + cos(a))*sin(a))
Parte trigonométrica [src] 
                           /       /    pi\\            
                           |    sec|x - --||            
           1               |       \    2 /|    /    pi\
------------------------ + |1 + -----------|*sec|a - --|
/      1   \    /    pi\   \       sec(x)  /    \    2 /
|1 + ------|*sec|a - --|                                
\    sec(a)/    \    2 /
$$\left(1 + \frac{\sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}}\right) \sec{\left(a - \frac{\pi}{2} \right)} + \frac{1}{\left(1 + \frac{1}{\sec{\left(a \right)}}\right) \sec{\left(a - \frac{\pi}{2} \right)}}$$ 
                     cos(x)  
              1 + -----------
   /    pi\          /    pi\
cos|a - --|       cos|x - --|
   \    2 /          \    2 /
----------- + ---------------
 1 + cos(a)        /    pi\  
                cos|a - --|  
                   \    2 /
$$\frac{\frac{\cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}} + 1}{\cos{\left(a - \frac{\pi}{2} \right)}} + \frac{\cos{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(a \right)} + 1}$$ 
                       sin(2*x)
                  1 + ---------
                           2   
     sin(a)           2*sin (x)
--------------- + -------------
       /    pi\       sin(a)   
1 + sin|a + --|                
       \    2 /
$$\frac{1 + \frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(x \right)}}}{\sin{\left(a \right)}} + \frac{\sin{\left(a \right)}}{\sin{\left(a + \frac{\pi}{2} \right)} + 1}$$ 
  sin(a)     /      1   \       
---------- + |1 + ------|*csc(a)
1 + cos(a)   \    tan(x)/
$$\left(1 + \frac{1}{\tan{\left(x \right)}}\right) \csc{\left(a \right)} + \frac{\sin{\left(a \right)}}{\cos{\left(a \right)} + 1}$$ 
         1            /    csc(x)\       
------------------- + |1 + ------|*csc(a)
/      1   \          \    sec(x)/       
|1 + ------|*csc(a)                      
\    sec(a)/
$$\left(\frac{\csc{\left(x \right)}}{\sec{\left(x \right)}} + 1\right) \csc{\left(a \right)} + \frac{1}{\left(1 + \frac{1}{\sec{\left(a \right)}}\right) \csc{\left(a \right)}}$$ 
/       2/a\\                                 /a\            
|1 + cot |-||*(1 + cot(x))               2*cot|-|            
\        \2//                                 \2/            
-------------------------- + --------------------------------
              /a\                          /            2/a\\
         2*cot|-|                          |    -1 + cot |-||
              \2/            /       2/a\\ |             \2/|
                             |1 + cot |-||*|1 + ------------|
                             \        \2// |           2/a\ |
                                           |    1 + cot |-| |
                                           \            \2/ /
$$\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\cot{\left(x \right)} + 1\right)}{2 \cot{\left(\frac{a}{2} \right)}} + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$ 
                  sin(2*x)
             1 + ---------
                      2   
  sin(a)         2*sin (x)
---------- + -------------
1 + cos(a)       sin(a)
$$\frac{1 + \frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(x \right)}}}{\sin{\left(a \right)}} + \frac{\sin{\left(a \right)}}{\cos{\left(a \right)} + 1}$$ 
                           /       /    pi\\            
                           |    sec|x - --||            
           1               |       \    2 /|    /pi    \
------------------------ + |1 + -----------|*sec|-- - a|
/      1   \    /    pi\   \       sec(x)  /    \2     /
|1 + ------|*sec|a - --|                                
\    sec(a)/    \    2 /
$$\left(1 + \frac{\sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}}\right) \sec{\left(- a + \frac{\pi}{2} \right)} + \frac{1}{\left(1 + \frac{1}{\sec{\left(a \right)}}\right) \sec{\left(a - \frac{\pi}{2} \right)}}$$ 
           1               /       csc(x)  \       
------------------------ + |1 + -----------|*csc(a)
/         1     \          |       /pi    \|       
|1 + -----------|*csc(a)   |    csc|-- - x||       
|       /pi    \|          \       \2     //       
|    csc|-- - a||                                  
\       \2     //
$$\left(\frac{\csc{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(a \right)} + \frac{1}{\left(1 + \frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}}\right) \csc{\left(a \right)}}$$ 
                 cos(x)
             1 + ------
  sin(a)         sin(x)
---------- + ----------
1 + cos(a)     sin(a)
$$\frac{1 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}}{\sin{\left(a \right)}} + \frac{\sin{\left(a \right)}}{\cos{\left(a \right)} + 1}$$ 
/      1   \ /       2/a\\                    /a\           
|1 + ------|*|1 + tan |-||               2*tan|-|           
\    tan(x)/ \        \2//                    \2/           
-------------------------- + -------------------------------
              /a\                          /           2/a\\
         2*tan|-|                          |    1 - tan |-||
              \2/            /       2/a\\ |            \2/|
                             |1 + tan |-||*|1 + -----------|
                             \        \2// |           2/a\|
                                           |    1 + tan |-||
                                           \            \2//
$$\frac{\left(1 + \frac{1}{\tan{\left(x \right)}}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}{2 \tan{\left(\frac{a}{2} \right)}} + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$ 
                   1   
             1 + ------
  sin(a)         tan(x)
---------- + ----------
1 + cos(a)     sin(a)
$$\frac{1 + \frac{1}{\tan{\left(x \right)}}}{\sin{\left(a \right)}} + \frac{\sin{\left(a \right)}}{\cos{\left(a \right)} + 1}$$ 
sin(a)/(1 + cos(a)) + (1 + 1/tan(x))/sin(a)
Denominador común [src] 
       2                                     
1 + sin (a) + cos(a)*cot(x) + cos(a) + cot(x)
---------------------------------------------
            cos(a)*sin(a) + sin(a)
$$\frac{\sin^{2}{\left(a \right)} + \cos{\left(a \right)} \cot{\left(x \right)} + \cos{\left(a \right)} + \cot{\left(x \right)} + 1}{\sin{\left(a \right)} \cos{\left(a \right)} + \sin{\left(a \right)}}$$ 
(1 + sin(a)^2 + cos(a)*cot(x) + cos(a) + cot(x))/(cos(a)*sin(a) + sin(a))
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