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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
  sin(a)     1 + cos(a)
---------- + ----------
1 + cos(a)     sin(a)
cos⁡(a)+1sin⁡(a)+sin⁡(a)cos⁡(a)+1\frac{\cos{\left(a \right)} + 1}{\sin{\left(a \right)}} + \frac{\sin{\left(a \right)}}{\cos{\left(a \right)} + 1} 
sin(a)/(1 + cos(a)) + (1 + cos(a))/sin(a)
Simplificación general [src] 
  2   
------
sin(a)
2sin⁡(a)\frac{2}{\sin{\left(a \right)}} 
2/sin(a)
Respuesta numérica [src] 
sin(a)/(1.0 + cos(a)) + (1.0 + cos(a))/sin(a)
sin(a)/(1.0 + cos(a)) + (1.0 + cos(a))/sin(a)
Denominador racional [src] 
            2      2   
(1 + cos(a))  + sin (a)
-----------------------
  (1 + cos(a))*sin(a)
(cos⁡(a)+1)2+sin⁡2(a)(cos⁡(a)+1)sin⁡(a)\frac{\left(\cos{\left(a \right)} + 1\right)^{2} + \sin^{2}{\left(a \right)}}{\left(\cos{\left(a \right)} + 1\right) \sin{\left(a \right)}} 
((1 + cos(a))^2 + sin(a)^2)/((1 + cos(a))*sin(a))
Denominador común [src] 
       2         2              
1 + cos (a) + sin (a) + 2*cos(a)
--------------------------------
     cos(a)*sin(a) + sin(a)
sin⁡2(a)+cos⁡2(a)+2cos⁡(a)+1sin⁡(a)cos⁡(a)+sin⁡(a)\frac{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)} + 2 \cos{\left(a \right)} + 1}{\sin{\left(a \right)} \cos{\left(a \right)} + \sin{\left(a \right)}} 
(1 + cos(a)^2 + sin(a)^2 + 2*cos(a))/(cos(a)*sin(a) + sin(a))
Potencias [src] 
    /     I*a    -I*a\                       
    |    e      e    |                       
2*I*|1 + ---- + -----|      /   -I*a    I*a\ 
    \     2       2  /    I*\- e     + e   / 
---------------------- - --------------------
       -I*a    I*a         /     I*a    -I*a\
    - e     + e            |    e      e    |
                         2*|1 + ---- + -----|
                           \     2       2  /
−i(eia−e−ia)2(eia2+1+e−ia2)+2i(eia2+1+e−ia2)eia−e−ia- \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(\frac{e^{i a}}{2} + 1 + \frac{e^{- i a}}{2}\right)}{e^{i a} - e^{- i a}} 
2*i*(1 + exp(i*a)/2 + exp(-i*a)/2)/(-exp(-i*a) + exp(i*a)) - i*(-exp(-i*a) + exp(i*a))/(2*(1 + exp(i*a)/2 + exp(-i*a)/2))
Unión de expresiones racionales [src] 
            2      2   
(1 + cos(a))  + sin (a)
-----------------------
  (1 + cos(a))*sin(a)
(cos⁡(a)+1)2+sin⁡2(a)(cos⁡(a)+1)sin⁡(a)\frac{\left(\cos{\left(a \right)} + 1\right)^{2} + \sin^{2}{\left(a \right)}}{\left(\cos{\left(a \right)} + 1\right) \sin{\left(a \right)}} 
((1 + cos(a))^2 + sin(a)^2)/((1 + cos(a))*sin(a))
Combinatoria [src] 
       2         2              
1 + cos (a) + sin (a) + 2*cos(a)
--------------------------------
      (1 + cos(a))*sin(a)
sin⁡2(a)+cos⁡2(a)+2cos⁡(a)+1(cos⁡(a)+1)sin⁡(a)\frac{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)} + 2 \cos{\left(a \right)} + 1}{\left(\cos{\left(a \right)} + 1\right) \sin{\left(a \right)}} 
(1 + cos(a)^2 + sin(a)^2 + 2*cos(a))/((1 + cos(a))*sin(a))
Abrimos la expresión [src] 
  1      cos(a)     sin(a)  
------ + ------ + ----------
sin(a)   sin(a)   1 + cos(a)
cos⁡(a)sin⁡(a)+1sin⁡(a)+sin⁡(a)cos⁡(a)+1\frac{\cos{\left(a \right)}}{\sin{\left(a \right)}} + \frac{1}{\sin{\left(a \right)}} + \frac{\sin{\left(a \right)}}{\cos{\left(a \right)} + 1} 
1/sin(a) + cos(a)/sin(a) + sin(a)/(1 + cos(a))
Parte trigonométrica [src] 
     2     
-----------
   /    pi\
cos|a - --|
   \    2 /
2cos⁡(a−π2)\frac{2}{\cos{\left(a - \frac{\pi}{2} \right)}} 
     /    pi\
2*sec|a - --|
     \    2 /
2sec⁡(a−π2)2 \sec{\left(a - \frac{\pi}{2} \right)} 
   /    pi\              
cos|a - --|              
   \    2 /    1 + cos(a)
----------- + -----------
 1 + cos(a)      /    pi\
              cos|a - --|
                 \    2 /
cos⁡(a)+1cos⁡(a−π2)+cos⁡(a−π2)cos⁡(a)+1\frac{\cos{\left(a \right)} + 1}{\cos{\left(a - \frac{\pi}{2} \right)}} + \frac{\cos{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(a \right)} + 1} 
           1               /      1   \    /    pi\
------------------------ + |1 + ------|*sec|a - --|
/      1   \    /    pi\   \    sec(a)/    \    2 /
|1 + ------|*sec|a - --|                           
\    sec(a)/    \    2 /
(1+1sec⁡(a))sec⁡(a−π2)+1(1+1sec⁡(a))sec⁡(a−π2)\left(1 + \frac{1}{\sec{\left(a \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)}} 
       2/a\
1 + tan |-|
        \2/
-----------
      /a\  
   tan|-|  
      \2/
tan⁡2(a2)+1tan⁡(a2)\frac{\tan^{2}{\left(\frac{a}{2} \right)} + 1}{\tan{\left(\frac{a}{2} \right)}} 
2*csc(a)
2csc⁡(a)2 \csc{\left(a \right)} 
         1            /      1   \       
------------------- + |1 + ------|*csc(a)
/      1   \          \    sec(a)/       
|1 + ------|*csc(a)                      
\    sec(a)/
(1+1sec⁡(a))csc⁡(a)+1(1+1sec⁡(a))csc⁡(a)\left(1 + \frac{1}{\sec{\left(a \right)}}\right) \csc{\left(a \right)} + \frac{1}{\left(1 + \frac{1}{\sec{\left(a \right)}}\right) \csc{\left(a \right)}} 
           1               /         1     \       
------------------------ + |1 + -----------|*csc(a)
/         1     \          |       /pi    \|       
|1 + -----------|*csc(a)   |    csc|-- - a||       
|       /pi    \|          \       \2     //       
|    csc|-- - a||                                  
\       \2     //
(1+1csc⁡(−a+π2))csc⁡(a)+1(1+1csc⁡(−a+π2))csc⁡(a)\left(1 + \frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}}\right) \csc{\left(a \right)} + \frac{1}{\left(1 + \frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}}\right) \csc{\left(a \right)}} 
                         /    pi\
                  1 + sin|a + --|
     sin(a)              \    2 /
--------------- + ---------------
       /    pi\        sin(a)    
1 + sin|a + --|                  
       \    2 /
sin⁡(a+π2)+1sin⁡(a)+sin⁡(a)sin⁡(a+π2)+1\frac{\sin{\left(a + \frac{\pi}{2} \right)} + 1}{\sin{\left(a \right)}} + \frac{\sin{\left(a \right)}}{\sin{\left(a + \frac{\pi}{2} \right)} + 1} 
              /           2/a\\                                  
              |    1 - tan |-||                                  
/       2/a\\ |            \2/|                                  
|1 + tan |-||*|1 + -----------|                                  
\        \2// |           2/a\|                    /a\           
              |    1 + tan |-||               2*tan|-|           
              \            \2//                    \2/           
------------------------------- + -------------------------------
                 /a\                            /           2/a\\
            2*tan|-|                            |    1 - tan |-||
                 \2/              /       2/a\\ |            \2/|
                                  |1 + tan |-||*|1 + -----------|
                                  \        \2// |           2/a\|
                                                |    1 + tan |-||
                                                \            \2//
(1−tan⁡2(a2)tan⁡2(a2)+1+1)(tan⁡2(a2)+1)2tan⁡(a2)+2tan⁡(a2)(1−tan⁡2(a2)tan⁡2(a2)+1+1)(tan⁡2(a2)+1)\frac{\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)}{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)} 
              /            2/a\\                                   
              |    -1 + cot |-||                                   
/       2/a\\ |             \2/|                                   
|1 + cot |-||*|1 + ------------|                                   
\        \2// |           2/a\ |                    /a\            
              |    1 + cot |-| |               2*cot|-|            
              \            \2/ /                    \2/            
-------------------------------- + --------------------------------
                 /a\                             /            2/a\\
            2*cot|-|                             |    -1 + cot |-||
                 \2/               /       2/a\\ |             \2/|
                                   |1 + cot |-||*|1 + ------------|
                                   \        \2// |           2/a\ |
                                                 |    1 + cot |-| |
                                                 \            \2/ /
(cot⁡2(a2)−1cot⁡2(a2)+1+1)(cot⁡2(a2)+1)2cot⁡(a2)+2cot⁡(a2)(cot⁡2(a2)−1cot⁡2(a2)+1+1)(cot⁡2(a2)+1)\frac{\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)}{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)} 
  2   
------
sin(a)
2sin⁡(a)\frac{2}{\sin{\left(a \right)}} 
       2/a\
1 + cot |-|
        \2/
-----------
      /a\  
   cot|-|  
      \2/
cot⁡2(a2)+1cot⁡(a2)\frac{\cot^{2}{\left(\frac{a}{2} \right)} + 1}{\cot{\left(\frac{a}{2} \right)}} 
(1 + cot(a/2)^2)/cot(a/2)
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