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¿Cómo vas a descomponer esta sin^2a/(cosa-1) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src]
    2     
 sin (a)  
----------
cos(a) - 1
$$\frac{\sin^{2}{\left(a \right)}}{\cos{\left(a \right)} - 1}$$
sin(a)^2/(cos(a) - 1)
Potencias [src]
                  2  
  /   -I*a    I*a\   
 -\- e     + e   /   
---------------------
  /      I*a    -I*a\
  |     e      e    |
4*|-1 + ---- + -----|
  \      2       2  /
$$- \frac{\left(e^{i a} - e^{- i a}\right)^{2}}{4 \left(\frac{e^{i a}}{2} - 1 + \frac{e^{- i a}}{2}\right)}$$
-(-exp(-i*a) + exp(i*a))^2/(4*(-1 + exp(i*a)/2 + exp(-i*a)/2))
Respuesta numérica [src]
sin(a)^2/(-1.0 + cos(a))
sin(a)^2/(-1.0 + cos(a))
Parte trigonométrica [src]
            1             
--------------------------
/       1   \    2/    pi\
|-1 + ------|*sec |a - --|
\     sec(a)/     \    2 /
$$\frac{1}{\left(-1 + \frac{1}{\sec{\left(a \right)}}\right) \sec^{2}{\left(a - \frac{\pi}{2} \right)}}$$
             2      
      1 - tan (a)   
  1 - -----------   
             2      
      1 + tan (a)   
--------------------
  /            2/a\\
  |     1 - tan |-||
  |             \2/|
2*|-1 + -----------|
  |            2/a\|
  |     1 + tan |-||
  \             \2//
$$\frac{- \frac{1 - \tan^{2}{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} + 1}{2 \left(\frac{1 - \tan^{2}{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} - 1\right)}$$
       2        
    sin (a)     
----------------
        /    pi\
-1 + sin|a + --|
        \    2 /
$$\frac{\sin^{2}{\left(a \right)}}{\sin{\left(a + \frac{\pi}{2} \right)} - 1}$$
   2/    pi\
cos |a - --|
    \    2 /
------------
-1 + cos(a) 
$$\frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(a \right)} - 1}$$
           1        
 1 - -------------  
        /pi      \  
     csc|-- - 2*a|  
        \2       /  
--------------------
  /          1     \
2*|-1 + -----------|
  |        /pi    \|
  |     csc|-- - a||
  \        \2     //
$$\frac{1 - \frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}}}{2 \left(-1 + \frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}}\right)}$$
                 2/a\            
            4*tan |-|            
                  \2/            
---------------------------------
               /            2/a\\
             2 |     1 - tan |-||
/       2/a\\  |             \2/|
|1 + tan |-|| *|-1 + -----------|
\        \2//  |            2/a\|
               |     1 + tan |-||
               \             \2//
$$\frac{4 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} - 1\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
                 2/a\             
            4*cot |-|             
                  \2/             
----------------------------------
               /             2/a\\
             2 |     -1 + cot |-||
/       2/a\\  |              \2/|
|1 + cot |-|| *|-1 + ------------|
\        \2//  |            2/a\ |
               |     1 + cot |-| |
               \             \2/ /
$$\frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} - 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
               2     
       -1 + cot (a)  
   1 - ------------  
              2      
       1 + cot (a)   
---------------------
  /             2/a\\
  |     -1 + cot |-||
  |              \2/|
2*|-1 + ------------|
  |            2/a\ |
  |     1 + cot |-| |
  \             \2/ /
$$\frac{- \frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} + 1}{2 \left(\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} - 1\right)}$$
            1             
--------------------------
/          1     \    2   
|-1 + -----------|*csc (a)
|        /pi    \|        
|     csc|-- - a||        
\        \2     //        
$$\frac{1}{\left(-1 + \frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}}\right) \csc^{2}{\left(a \right)}}$$
          1          
---------------------
/       1   \    2   
|-1 + ------|*csc (a)
\     sec(a)/        
$$\frac{1}{\left(-1 + \frac{1}{\sec{\left(a \right)}}\right) \csc^{2}{\left(a \right)}}$$
        /pi      \  
 1 - sin|-- + 2*a|  
        \2       /  
--------------------
  /        /    pi\\
2*|-1 + sin|a + --||
  \        \    2 //
$$\frac{1 - \sin{\left(2 a + \frac{\pi}{2} \right)}}{2 \left(\sin{\left(a + \frac{\pi}{2} \right)} - 1\right)}$$
         1     
  1 - -------- 
      sec(2*a) 
---------------
  /       1   \
2*|-1 + ------|
  \     sec(a)/
$$\frac{1 - \frac{1}{\sec{\left(2 a \right)}}}{2 \left(-1 + \frac{1}{\sec{\left(a \right)}}\right)}$$
  1 - cos(2*a) 
---------------
2*(-1 + cos(a))
$$\frac{1 - \cos{\left(2 a \right)}}{2 \left(\cos{\left(a \right)} - 1\right)}$$
(1 - cos(2*a))/(2*(-1 + cos(a)))