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Simplificación de expresiones/ Descomposición en fracciones simples/ cos2*a/(1-sin*a)*(1+sin*a)

¿Cómo vas a descomponer esta cos2*a/(1-sin*a)*(1+sin*a) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
 cos(2*a)              
----------*(1 + sin(a))
1 - sin(a)
$$\frac{\cos{\left(2 a \right)}}{1 - \sin{\left(a \right)}} \left(\sin{\left(a \right)} + 1\right)$$ 
(cos(2*a)/(1 - sin(a)))*(1 + sin(a))
Simplificación general [src] 
-(1 + sin(a))*cos(2*a) 
-----------------------
      -1 + sin(a)
$$- \frac{\left(\sin{\left(a \right)} + 1\right) \cos{\left(2 a \right)}}{\sin{\left(a \right)} - 1}$$ 
-(1 + sin(a))*cos(2*a)/(-1 + sin(a))
Potencias [src] 
/      /   -I*a    I*a\\ / -2*I*a    2*I*a\
|    I*\- e     + e   /| |e         e     |
|1 - ------------------|*|------- + ------|
\            2         / \   2        2   /
-------------------------------------------
                 /   -I*a    I*a\          
               I*\- e     + e   /          
           1 + ------------------          
                       2
$$\frac{\left(- \frac{i \left(e^{i a} - e^{- i a}\right)}{2} + 1\right) \left(\frac{e^{2 i a}}{2} + \frac{e^{- 2 i a}}{2}\right)}{\frac{i \left(e^{i a} - e^{- i a}\right)}{2} + 1}$$ 
(1 + sin(a))*cos(2*a)
---------------------
      1 - sin(a)
$$\frac{\left(\sin{\left(a \right)} + 1\right) \cos{\left(2 a \right)}}{1 - \sin{\left(a \right)}}$$ 
(1 + sin(a))*cos(2*a)/(1 - sin(a))
Respuesta numérica [src] 
(1.0 + sin(a))*cos(2*a)/(1.0 - sin(a))
(1.0 + sin(a))*cos(2*a)/(1.0 - sin(a))
Denominador común [src] 
             2*cos(2*a)
-cos(2*a) - -----------
            -1 + sin(a)
$$- \cos{\left(2 a \right)} - \frac{2 \cos{\left(2 a \right)}}{\sin{\left(a \right)} - 1}$$ 
-cos(2*a) - 2*cos(2*a)/(-1 + sin(a))
Denominador racional [src] 
(1 + sin(a))*cos(2*a)
---------------------
      1 - sin(a)
$$\frac{\left(\sin{\left(a \right)} + 1\right) \cos{\left(2 a \right)}}{1 - \sin{\left(a \right)}}$$ 
(1 + sin(a))*cos(2*a)/(1 - sin(a))
Unión de expresiones racionales [src] 
(1 + sin(a))*cos(2*a)
---------------------
      1 - sin(a)
$$\frac{\left(\sin{\left(a \right)} + 1\right) \cos{\left(2 a \right)}}{1 - \sin{\left(a \right)}}$$ 
(1 + sin(a))*cos(2*a)/(1 - sin(a))
Abrimos la expresión [src] 
                                 2            2          
      1          sin(a)     2*cos (a)    2*cos (a)*sin(a)
- ---------- - ---------- + ---------- + ----------------
  1 - sin(a)   1 - sin(a)   1 - sin(a)      1 - sin(a)
$$\frac{2 \sin{\left(a \right)} \cos^{2}{\left(a \right)}}{1 - \sin{\left(a \right)}} - \frac{\sin{\left(a \right)}}{1 - \sin{\left(a \right)}} + \frac{2 \cos^{2}{\left(a \right)}}{1 - \sin{\left(a \right)}} - \frac{1}{1 - \sin{\left(a \right)}}$$ 
-1/(1 - sin(a)) - sin(a)/(1 - sin(a)) + 2*cos(a)^2/(1 - sin(a)) + 2*cos(a)^2*sin(a)/(1 - sin(a))
Compilar la expresión [src] 
(1 + sin(a))*cos(2*a)
---------------------
      1 - sin(a)
$$\frac{\left(\sin{\left(a \right)} + 1\right) \cos{\left(2 a \right)}}{1 - \sin{\left(a \right)}}$$ 
(1 + sin(a))*cos(2*a)/(1 - sin(a))
Combinatoria [src] 
-(1 + sin(a))*cos(2*a) 
-----------------------
      -1 + sin(a)
$$- \frac{\left(\sin{\left(a \right)} + 1\right) \cos{\left(2 a \right)}}{\sin{\left(a \right)} - 1}$$ 
-(1 + sin(a))*cos(2*a)/(-1 + sin(a))
Parte trigonométrica [src] 
              /         4   \
            2 |    4*sin (a)|
(1 + sin(a)) *|1 - ---------|
              |       2     |
              \    sin (2*a)/
$$\left(- \frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\left(2 a \right)}} + 1\right) \left(\sin{\left(a \right)} + 1\right)^{2}$$ 
            2 /       2   \
(1 + sin(a)) *\1 - tan (a)/
$$\left(1 - \tan^{2}{\left(a \right)}\right) \left(\sin{\left(a \right)} + 1\right)^{2}$$ 
              /       2/pi    \\
            2 |    csc |-- - a||
/      1   \  |        \2     /|
|1 + ------| *|1 - ------------|
\    csc(a)/  |         2      |
              \      csc (a)   /
$$\left(1 - \frac{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(a \right)}}\right) \left(1 + \frac{1}{\csc{\left(a \right)}}\right)^{2}$$ 
          /           /a\ \               
          |      2*tan|-| |               
     2/a\ |           \2/ | /        1   \
4*tan |-|*|1 + -----------|*|-1 + -------|
      \2/ |           2/a\| |        2   |
          |    1 + tan |-|| \     tan (a)/
          \            \2//               
------------------------------------------
                    /           /a\ \     
                  2 |      2*tan|-| |     
     /       2/a\\  |           \2/ |     
     |1 + tan |-|| *|1 - -----------|     
     \        \2//  |           2/a\|     
                    |    1 + tan |-||     
                    \            \2//
$$\frac{4 \left(-1 + \frac{1}{\tan^{2}{\left(a \right)}}\right) \left(1 + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \tan^{2}{\left(\frac{a}{2} \right)}}{\left(1 - \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$ 
     4/a\    2/a\              /        1   \
4*cos |-|*tan |-|*(1 + sin(a))*|-1 + -------|
      \2/     \2/              |        2   |
                               \     tan (a)/
---------------------------------------------
                  1 - sin(a)
$$\frac{4 \left(-1 + \frac{1}{\tan^{2}{\left(a \right)}}\right) \left(\sin{\left(a \right)} + 1\right) \cos^{4}{\left(\frac{a}{2} \right)} \tan^{2}{\left(\frac{a}{2} \right)}}{1 - \sin{\left(a \right)}}$$ 
   2                 /        2   \
sin (a)*(1 + sin(a))*\-1 + cot (a)/
-----------------------------------
             1 - sin(a)
$$\frac{\left(\sin{\left(a \right)} + 1\right) \left(\cot^{2}{\left(a \right)} - 1\right) \sin^{2}{\left(a \right)}}{1 - \sin{\left(a \right)}}$$ 
          /           /a\ \               
          |      2*cot|-| |               
     2/a\ |           \2/ | /        2   \
4*cot |-|*|1 + -----------|*\-1 + cot (a)/
      \2/ |           2/a\|               
          |    1 + cot |-||               
          \            \2//               
------------------------------------------
                    /           /a\ \     
                  2 |      2*cot|-| |     
     /       2/a\\  |           \2/ |     
     |1 + cot |-|| *|1 - -----------|     
     \        \2//  |           2/a\|     
                    |    1 + cot |-||     
                    \            \2//
$$\frac{4 \left(1 + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\cot^{2}{\left(a \right)} - 1\right) \cot^{2}{\left(\frac{a}{2} \right)}}{\left(1 - \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$ 
(1 + sin(a))*cos(2*a)
---------------------
      1 - sin(a)
$$\frac{\left(\sin{\left(a \right)} + 1\right) \cos{\left(2 a \right)}}{1 - \sin{\left(a \right)}}$$ 
                /pi      \
(1 + sin(a))*sin|-- + 2*a|
                \2       /
--------------------------
        1 - sin(a)
$$\frac{\left(\sin{\left(a \right)} + 1\right) \sin{\left(2 a + \frac{\pi}{2} \right)}}{1 - \sin{\left(a \right)}}$$ 
                  /        2/    pi\\
                  |     sec |a - --||
/         1     \ |         \    2 /|
|1 + -----------|*|-1 + ------------|
|       /    pi\| |          2      |
|    sec|a - --|| \       sec (a)   /
\       \    2 //                    
-------------------------------------
    /         1     \    2/    pi\   
    |1 - -----------|*sec |a - --|   
    |       /    pi\|     \    2 /   
    |    sec|a - --||                
    \       \    2 //
$$\frac{\left(-1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}\right) \left(1 + \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}\right)}{\left(1 - \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}\right) \sec^{2}{\left(a - \frac{\pi}{2} \right)}}$$ 
              1           
     1 + -----------      
            /    pi\      
         sec|a - --|      
            \    2 /      
--------------------------
/         1     \         
|1 - -----------|*sec(2*a)
|       /    pi\|         
|    sec|a - --||         
\       \    2 //
$$\frac{1 + \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}}{\left(1 - \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}\right) \sec{\left(2 a \right)}}$$ 
                     /        2     \
   2                 |     sin (2*a)|
sin (a)*(1 + sin(a))*|-1 + ---------|
                     |          4   |
                     \     4*sin (a)/
-------------------------------------
              1 - sin(a)
$$\frac{\left(-1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}\right) \left(\sin{\left(a \right)} + 1\right) \sin^{2}{\left(a \right)}}{1 - \sin{\left(a \right)}}$$ 
     2/a\    4/a\              /        2   \
4*cot |-|*sin |-|*(1 + sin(a))*\-1 + cot (a)/
      \2/     \2/                            
---------------------------------------------
                  1 - sin(a)
$$\frac{4 \left(\sin{\left(a \right)} + 1\right) \left(\cot^{2}{\left(a \right)} - 1\right) \sin^{4}{\left(\frac{a}{2} \right)} \cot^{2}{\left(\frac{a}{2} \right)}}{1 - \sin{\left(a \right)}}$$ 
              1           
        1 + ------        
            csc(a)        
--------------------------
/      1   \    /pi      \
|1 - ------|*csc|-- - 2*a|
\    csc(a)/    \2       /
$$\frac{1 + \frac{1}{\csc{\left(a \right)}}}{\left(1 - \frac{1}{\csc{\left(a \right)}}\right) \csc{\left(- 2 a + \frac{\pi}{2} \right)}}$$ 
                 2              
/           /a\ \               
|      2*tan|-| |               
|           \2/ |  /       2   \
|1 + -----------| *\1 - tan (a)/
|           2/a\|               
|    1 + tan |-||               
\            \2//
$$\left(1 + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right)^{2} \left(1 - \tan^{2}{\left(a \right)}\right)$$ 
/       /    pi\\         
|1 + cos|a - --||*cos(2*a)
\       \    2 //         
--------------------------
            /    pi\      
     1 - cos|a - --|      
            \    2 /
$$\frac{\left(\cos{\left(a - \frac{\pi}{2} \right)} + 1\right) \cos{\left(2 a \right)}}{1 - \cos{\left(a - \frac{\pi}{2} \right)}}$$ 
             /          2      \
/      1   \ |       csc (a)   |
|1 + ------|*|-1 + ------------|
\    csc(a)/ |        2/pi    \|
             |     csc |-- - a||
             \         \2     //
--------------------------------
      /      1   \    2         
      |1 - ------|*csc (a)      
      \    csc(a)/
$$\frac{\left(1 + \frac{1}{\csc{\left(a \right)}}\right) \left(\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} - 1\right)}{\left(1 - \frac{1}{\csc{\left(a \right)}}\right) \csc^{2}{\left(a \right)}}$$ 
            1        
      1 + ------     
          csc(a)     
---------------------
/      1   \         
|1 - ------|*sec(2*a)
\    csc(a)/
$$\frac{1 + \frac{1}{\csc{\left(a \right)}}}{\left(1 - \frac{1}{\csc{\left(a \right)}}\right) \sec{\left(2 a \right)}}$$ 
              /           /a\ \
              |      2*tan|-| |
/       2   \ |           \2/ |
\1 - tan (a)/*|1 + -----------|
              |           2/a\|
              |    1 + tan |-||
              \            \2//
-------------------------------
              /           /a\ \
              |      2*tan|-| |
/       2   \ |           \2/ |
\1 + tan (a)/*|1 - -----------|
              |           2/a\|
              |    1 + tan |-||
              \            \2//
$$\frac{\left(1 + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(1 - \tan^{2}{\left(a \right)}\right)}{\left(1 - \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(a \right)} + 1\right)}$$ 
                   /       2/    pi\\
                 2 |    cos |a - --||
/       /    pi\\  |        \    2 /|
|1 + cos|a - --|| *|1 - ------------|
\       \    2 //  |         2      |
                   \      cos (a)   /
$$\left(1 - \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(a \right)}}\right) \left(\cos{\left(a - \frac{\pi}{2} \right)} + 1\right)^{2}$$ 
/           /a\ \               
|      2*cot|-| |               
|           \2/ | /        2   \
|1 + -----------|*\-1 + cot (a)/
|           2/a\|               
|    1 + cot |-||               
\            \2//               
--------------------------------
              /           /a\ \ 
              |      2*cot|-| | 
/       2   \ |           \2/ | 
\1 + cot (a)/*|1 - -----------| 
              |           2/a\| 
              |    1 + cot |-|| 
              \            \2//
$$\frac{\left(1 + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\cot^{2}{\left(a \right)} - 1\right)}{\left(1 - \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\cot^{2}{\left(a \right)} + 1\right)}$$ 
                 2 /         2      \
/         1     \  |      sec (a)   |
|1 + -----------| *|1 - ------------|
|       /    pi\|  |       2/    pi\|
|    sec|a - --||  |    sec |a - --||
\       \    2 //  \        \    2 //
$$\left(1 + \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}\right)^{2} \left(- \frac{\sec^{2}{\left(a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + 1\right)$$ 
                               /          2      \
   2/    pi\ /       /    pi\\ |       cos (a)   |
cos |a - --|*|1 + cos|a - --||*|-1 + ------------|
    \    2 / \       \    2 // |        2/    pi\|
                               |     cos |a - --||
                               \         \    2 //
--------------------------------------------------
                        /    pi\                  
                 1 - cos|a - --|                  
                        \    2 /
$$\frac{\left(\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} - 1\right) \left(\cos{\left(a - \frac{\pi}{2} \right)} + 1\right) \cos^{2}{\left(a - \frac{\pi}{2} \right)}}{1 - \cos{\left(a - \frac{\pi}{2} \right)}}$$ 
                 2              
/           /a\ \               
|      2*cot|-| |               
|           \2/ |  /       1   \
|1 + -----------| *|1 - -------|
|           2/a\|  |       2   |
|    1 + cot |-||  \    cot (a)/
\            \2//
$$\left(1 + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right)^{2} \left(1 - \frac{1}{\cot^{2}{\left(a \right)}}\right)$$ 
(1 + 2*cot(a/2)/(1 + cot(a/2)^2))^2*(1 - 1/cot(a)^2)
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