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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
  sin(a)     cos(a)
---------- - ------
1 - sin(a)   sin(a)
−cos⁡(a)sin⁡(a)+sin⁡(a)1−sin⁡(a)- \frac{\cos{\left(a \right)}}{\sin{\left(a \right)}} + \frac{\sin{\left(a \right)}}{1 - \sin{\left(a \right)}} 
sin(a)/(1 - sin(a)) - cos(a)/sin(a)
Simplificación general [src] 
     sin(a)     cos(a)
- ----------- - ------
  -1 + sin(a)   sin(a)
−cos⁡(a)sin⁡(a)−sin⁡(a)sin⁡(a)−1- \frac{\cos{\left(a \right)}}{\sin{\left(a \right)}} - \frac{\sin{\left(a \right)}}{\sin{\left(a \right)} - 1} 
-sin(a)/(-1 + sin(a)) - cos(a)/sin(a)
Respuesta numérica [src] 
sin(a)/(1.0 - sin(a)) - cos(a)/sin(a)
sin(a)/(1.0 - sin(a)) - cos(a)/sin(a)
Denominador racional [src] 
   2                         
sin (a) - (1 - sin(a))*cos(a)
-----------------------------
     (1 - sin(a))*sin(a)
−(1−sin⁡(a))cos⁡(a)+sin⁡2(a)(1−sin⁡(a))sin⁡(a)\frac{- \left(1 - \sin{\left(a \right)}\right) \cos{\left(a \right)} + \sin^{2}{\left(a \right)}}{\left(1 - \sin{\left(a \right)}\right) \sin{\left(a \right)}} 
(sin(a)^2 - (1 - sin(a))*cos(a))/((1 - sin(a))*sin(a))
Potencias [src] 
      / I*a    -I*a\                             
      |e      e    |                             
  2*I*|---- + -----|         /   -I*a    I*a\    
      \ 2       2  /       I*\- e     + e   /    
- ------------------ - --------------------------
       -I*a    I*a       /      /   -I*a    I*a\\
    - e     + e          |    I*\- e     + e   /|
                       2*|1 + ------------------|
                         \            2         /
−2i(eia2+e−ia2)eia−e−ia−i(eia−e−ia)2(i(eia−e−ia)2+1)- \frac{2 i \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)}{e^{i a} - e^{- i a}} - \frac{i \left(e^{i a} - e^{- i a}\right)}{2 \left(\frac{i \left(e^{i a} - e^{- i a}\right)}{2} + 1\right)} 
-2*i*(exp(i*a)/2 + exp(-i*a)/2)/(-exp(-i*a) + exp(i*a)) - i*(-exp(-i*a) + exp(i*a))/(2*(1 + i*(-exp(-i*a) + exp(i*a))/2))
Combinatoria [src] 
 /   2                            \ 
-\sin (a) - cos(a) + cos(a)*sin(a)/ 
------------------------------------
        (-1 + sin(a))*sin(a)
−sin⁡2(a)+sin⁡(a)cos⁡(a)−cos⁡(a)(sin⁡(a)−1)sin⁡(a)- \frac{\sin^{2}{\left(a \right)} + \sin{\left(a \right)} \cos{\left(a \right)} - \cos{\left(a \right)}}{\left(\sin{\left(a \right)} - 1\right) \sin{\left(a \right)}} 
-(sin(a)^2 - cos(a) + cos(a)*sin(a))/((-1 + sin(a))*sin(a))
Unión de expresiones racionales [src] 
   2                         
sin (a) - (1 - sin(a))*cos(a)
-----------------------------
     (1 - sin(a))*sin(a)
−(1−sin⁡(a))cos⁡(a)+sin⁡2(a)(1−sin⁡(a))sin⁡(a)\frac{- \left(1 - \sin{\left(a \right)}\right) \cos{\left(a \right)} + \sin^{2}{\left(a \right)}}{\left(1 - \sin{\left(a \right)}\right) \sin{\left(a \right)}} 
(sin(a)^2 - (1 - sin(a))*cos(a))/((1 - sin(a))*sin(a))
Denominador común [src] 
     -cos(a) + cos(a)*sin(a) + sin(a)
-1 - --------------------------------
                2                    
             sin (a) - sin(a)
−1−sin⁡(a)cos⁡(a)+sin⁡(a)−cos⁡(a)sin⁡2(a)−sin⁡(a)-1 - \frac{\sin{\left(a \right)} \cos{\left(a \right)} + \sin{\left(a \right)} - \cos{\left(a \right)}}{\sin^{2}{\left(a \right)} - \sin{\left(a \right)}} 
-1 - (-cos(a) + cos(a)*sin(a) + sin(a))/(sin(a)^2 - sin(a))
Parte trigonométrica [src] 
   /a\                         
tan|-|                         
   \2/      1          sin(a)  
------ - -------- - -----------
  2           /a\   -1 + sin(a)
         2*tan|-|              
              \2/
tan⁡(a2)2−12tan⁡(a2)−sin⁡(a)sin⁡(a)−1\frac{\tan{\left(\frac{a}{2} \right)}}{2} - \frac{1}{2 \tan{\left(\frac{a}{2} \right)}} - \frac{\sin{\left(a \right)}}{\sin{\left(a \right)} - 1} 
   /a\                               /a\            
tan|-|                          2*tan|-|            
   \2/      1                        \2/            
------ - -------- - --------------------------------
  2           /a\                 /            /a\ \
         2*tan|-|                 |       2*tan|-| |
              \2/   /       2/a\\ |            \2/ |
                    |1 + tan |-||*|-1 + -----------|
                    \        \2// |            2/a\|
                                  |     1 + tan |-||
                                  \             \2//
tan⁡(a2)2−12tan⁡(a2)−2tan⁡(a2)(−1+2tan⁡(a2)tan⁡2(a2)+1)(tan⁡2(a2)+1)\frac{\tan{\left(\frac{a}{2} \right)}}{2} - \frac{1}{2 \tan{\left(\frac{a}{2} \right)}} - \frac{2 \tan{\left(\frac{a}{2} \right)}}{\left(-1 + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)} 
             sin(a)  
-cot(a) - -----------
          -1 + sin(a)
−cot⁡(a)−sin⁡(a)sin⁡(a)−1- \cot{\left(a \right)} - \frac{\sin{\left(a \right)}}{\sin{\left(a \right)} - 1} 
     /    pi\                
  cos|a - --|                
     \    2 /        cos(a)  
--------------- - -----------
       /    pi\      /    pi\
1 - cos|a - --|   cos|a - --|
       \    2 /      \    2 /
−cos⁡(a)cos⁡(a−π2)+cos⁡(a−π2)1−cos⁡(a−π2)- \frac{\cos{\left(a \right)}}{\cos{\left(a - \frac{\pi}{2} \right)}} + \frac{\cos{\left(a - \frac{\pi}{2} \right)}}{1 - \cos{\left(a - \frac{\pi}{2} \right)}} 
         1               csc(a)  
------------------- - -----------
/      1   \             /pi    \
|1 - ------|*csc(a)   csc|-- - a|
\    csc(a)/             \2     /
−csc⁡(a)csc⁡(−a+π2)+1(1−1csc⁡(a))csc⁡(a)- \frac{\csc{\left(a \right)}}{\csc{\left(- a + \frac{\pi}{2} \right)}} + \frac{1}{\left(1 - \frac{1}{\csc{\left(a \right)}}\right) \csc{\left(a \right)}} 
   /a\      /a\              
tan|-|   cot|-|              
   \2/      \2/      sin(a)  
------ - ------ - -----------
  2        2      -1 + sin(a)
tan⁡(a2)2−cot⁡(a2)2−sin⁡(a)sin⁡(a)−1\frac{\tan{\left(\frac{a}{2} \right)}}{2} - \frac{\cot{\left(\frac{a}{2} \right)}}{2} - \frac{\sin{\left(a \right)}}{\sin{\left(a \right)} - 1} 
   2/a\                          
sin |-|                          
    \2/      sin(a)       sin(a) 
------- - ----------- - ---------
 sin(a)   -1 + sin(a)        2/a\
                        4*sin |-|
                              \2/
sin⁡2(a2)sin⁡(a)−sin⁡(a)4sin⁡2(a2)−sin⁡(a)sin⁡(a)−1\frac{\sin^{2}{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)}} - \frac{\sin{\left(a \right)}}{4 \sin^{2}{\left(\frac{a}{2} \right)}} - \frac{\sin{\left(a \right)}}{\sin{\left(a \right)} - 1} 
  sin(a)      sin(2*a)
---------- - ---------
1 - sin(a)        2   
             2*sin (a)
−sin⁡(2a)2sin⁡2(a)+sin⁡(a)1−sin⁡(a)- \frac{\sin{\left(2 a \right)}}{2 \sin^{2}{\left(a \right)}} + \frac{\sin{\left(a \right)}}{1 - \sin{\left(a \right)}} 
                /    pi\
             sin|a + --|
  sin(a)        \    2 /
---------- - -----------
1 - sin(a)      sin(a)
−sin⁡(a+π2)sin⁡(a)+sin⁡(a)1−sin⁡(a)- \frac{\sin{\left(a + \frac{\pi}{2} \right)}}{\sin{\left(a \right)}} + \frac{\sin{\left(a \right)}}{1 - \sin{\left(a \right)}} 
   /pi   a\                                 /a\   
csc|-- - -|                              csc|-|   
   \2    2/            1                    \2/   
----------- - -------------------- - -------------
       /a\    /       1   \               /pi   a\
  2*csc|-|    |-1 + ------|*csc(a)   2*csc|-- - -|
       \2/    \     csc(a)/               \2    2/
−csc⁡(a2)2csc⁡(−a2+π2)+csc⁡(−a2+π2)2csc⁡(a2)−1(−1+1csc⁡(a))csc⁡(a)- \frac{\csc{\left(\frac{a}{2} \right)}}{2 \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + \frac{\csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{2 \csc{\left(\frac{a}{2} \right)}} - \frac{1}{\left(-1 + \frac{1}{\csc{\left(a \right)}}\right) \csc{\left(a \right)}} 
              /a\                    /a\            
           cot|-|               2*cot|-|            
   1          \2/                    \2/            
-------- - ------ - --------------------------------
     /a\     2                    /            /a\ \
2*cot|-|                          |       2*cot|-| |
     \2/            /       2/a\\ |            \2/ |
                    |1 + cot |-||*|-1 + -----------|
                    \        \2// |            2/a\|
                                  |     1 + cot |-||
                                  \             \2//
−cot⁡(a2)2+12cot⁡(a2)−2cot⁡(a2)(−1+2cot⁡(a2)cot⁡2(a2)+1)(cot⁡2(a2)+1)- \frac{\cot{\left(\frac{a}{2} \right)}}{2} + \frac{1}{2 \cot{\left(\frac{a}{2} \right)}} - \frac{2 \cot{\left(\frac{a}{2} \right)}}{\left(-1 + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)} 
   /a   pi\        /    pi\             /a\   
cos|- - --|     cos|a - --|          cos|-|   
   \2   2 /        \    2 /             \2/   
----------- - ---------------- - -------------
       /a\            /    pi\        /a   pi\
  2*cos|-|    -1 + cos|a - --|   2*cos|- - --|
       \2/            \    2 /        \2   2 /
−cos⁡(a2)2cos⁡(a2−π2)+cos⁡(a2−π2)2cos⁡(a2)−cos⁡(a−π2)cos⁡(a−π2)−1- \frac{\cos{\left(\frac{a}{2} \right)}}{2 \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + \frac{\cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{2 \cos{\left(\frac{a}{2} \right)}} - \frac{\cos{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(a - \frac{\pi}{2} \right)} - 1} 
   2/a\                               
sin |-|                               
    \2/      sin(a)         sin(a)    
------- - ----------- - --------------
 sin(a)   -1 + sin(a)   2*(1 - cos(a))
sin⁡2(a2)sin⁡(a)−sin⁡(a)sin⁡(a)−1−sin⁡(a)2(1−cos⁡(a))\frac{\sin^{2}{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)}} - \frac{\sin{\left(a \right)}}{\sin{\left(a \right)} - 1} - \frac{\sin{\left(a \right)}}{2 \left(1 - \cos{\left(a \right)}\right)} 
                                   /    pi\
                                sec|a - --|
              1                    \    2 /
----------------------------- - -----------
/         1     \    /    pi\      sec(a)  
|1 - -----------|*sec|a - --|              
|       /    pi\|    \    2 /              
|    sec|a - --||                          
\       \    2 //
−sec⁡(a−π2)sec⁡(a)+1(1−1sec⁡(a−π2))sec⁡(a−π2)- \frac{\sec{\left(a - \frac{\pi}{2} \right)}}{\sec{\left(a \right)}} + \frac{1}{\left(1 - \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}\right) \sec{\left(a - \frac{\pi}{2} \right)}} 
         1            csc(a)
------------------- - ------
/      1   \          sec(a)
|1 - ------|*csc(a)         
\    csc(a)/
−csc⁡(a)sec⁡(a)+1(1−1csc⁡(a))csc⁡(a)- \frac{\csc{\left(a \right)}}{\sec{\left(a \right)}} + \frac{1}{\left(1 - \frac{1}{\csc{\left(a \right)}}\right) \csc{\left(a \right)}} 
    1        sin(a)  
- ------ + ----------
  tan(a)   1 - sin(a)
−1tan⁡(a)+sin⁡(a)1−sin⁡(a)- \frac{1}{\tan{\left(a \right)}} + \frac{\sin{\left(a \right)}}{1 - \sin{\left(a \right)}} 
    1         sin(a)  
- ------ - -----------
  tan(a)   -1 + sin(a)
−1tan⁡(a)−sin⁡(a)sin⁡(a)−1- \frac{1}{\tan{\left(a \right)}} - \frac{\sin{\left(a \right)}}{\sin{\left(a \right)} - 1} 
          2/a\                    /a\           
  -1 + cot |-|               2*cot|-|           
           \2/                    \2/           
- ------------ + -------------------------------
         /a\                   /           /a\ \
    2*cot|-|                   |      2*cot|-| |
         \2/     /       2/a\\ |           \2/ |
                 |1 + cot |-||*|1 - -----------|
                 \        \2// |           2/a\|
                               |    1 + cot |-||
                               \            \2//
−cot⁡2(a2)−12cot⁡(a2)+2cot⁡(a2)(1−2cot⁡(a2)cot⁡2(a2)+1)(cot⁡2(a2)+1)- \frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{2 \cot{\left(\frac{a}{2} \right)}} + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\left(1 - \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)} 
                            /a\           
                       2*tan|-|           
    1                       \2/           
- ------ + -------------------------------
  tan(a)                 /           /a\ \
                         |      2*tan|-| |
           /       2/a\\ |           \2/ |
           |1 + tan |-||*|1 - -----------|
           \        \2// |           2/a\|
                         |    1 + tan |-||
                         \            \2//
−1tan⁡(a)+2tan⁡(a2)(1−2tan⁡(a2)tan⁡2(a2)+1)(tan⁡2(a2)+1)- \frac{1}{\tan{\left(a \right)}} + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\left(1 - \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)} 
       /a\                                          /a   pi\
    sec|-|                                       sec|- - --|
       \2/                    1                     \2   2 /
------------- - ------------------------------ - -----------
     /a   pi\   /          1     \    /    pi\          /a\ 
2*sec|- - --|   |-1 + -----------|*sec|a - --|     2*sec|-| 
     \2   2 /   |        /    pi\|    \    2 /          \2/ 
                |     sec|a - --||                          
                \        \    2 //
sec⁡(a2)2sec⁡(a2−π2)−sec⁡(a2−π2)2sec⁡(a2)−1(−1+1sec⁡(a−π2))sec⁡(a−π2)\frac{\sec{\left(\frac{a}{2} \right)}}{2 \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - \frac{\sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{2 \sec{\left(\frac{a}{2} \right)}} - \frac{1}{\left(-1 + \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}\right) \sec{\left(a - \frac{\pi}{2} \right)}} 
                           /a\           
                      2*cot|-|           
                           \2/           
-cot(a) + -------------------------------
                        /           /a\ \
                        |      2*cot|-| |
          /       2/a\\ |           \2/ |
          |1 + cot |-||*|1 - -----------|
          \        \2// |           2/a\|
                        |    1 + cot |-||
                        \            \2//
−cot⁡(a)+2cot⁡(a2)(1−2cot⁡(a2)cot⁡2(a2)+1)(cot⁡2(a2)+1)- \cot{\left(a \right)} + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\left(1 - \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)} 
         2/a\                    /a\           
  1 - tan |-|               2*tan|-|           
          \2/                    \2/           
- ----------- + -------------------------------
         /a\                  /           /a\ \
    2*tan|-|                  |      2*tan|-| |
         \2/    /       2/a\\ |           \2/ |
                |1 + tan |-||*|1 - -----------|
                \        \2// |           2/a\|
                              |    1 + tan |-||
                              \            \2//
−1−tan⁡2(a2)2tan⁡(a2)+2tan⁡(a2)(1−2tan⁡(a2)tan⁡2(a2)+1)(tan⁡2(a2)+1)- \frac{1 - \tan^{2}{\left(\frac{a}{2} \right)}}{2 \tan{\left(\frac{a}{2} \right)}} + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\left(1 - \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)} 
-(1 - tan(a/2)^2)/(2*tan(a/2)) + 2*tan(a/2)/((1 + tan(a/2)^2)*(1 - 2*tan(a/2)/(1 + tan(a/2)^2)))
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