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Simplificación de expresiones/ Descomposición en fracciones simples/ tg(-a)cos(-a)/(1+ctg^2(-a))

¿Cómo vas a descomponer esta tg(-a)cos(-a)/(1+ctg^2(-a)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
tan(-a)*cos(-a)
---------------
         2     
  1 + cot (-a)
$$\frac{\cos{\left(- a \right)} \tan{\left(- a \right)}}{\cot^{2}{\left(- a \right)} + 1}$$ 
(tan(-a)*cos(-a))/(1 + cot(-a)^2)
Simplificación general [src] 
    3   
-sin (a)
$$- \sin^{3}{\left(a \right)}$$ 
-sin(a)^3
Respuesta numérica [src] 
cos(-a)*tan(-a)/(1.0 + cot(-a)^2)
cos(-a)*tan(-a)/(1.0 + cot(-a)^2)
Combinatoria [src] 
-cos(a)*tan(a) 
---------------
         2     
  1 + cot (a)
$$- \frac{\cos{\left(a \right)} \tan{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1}$$ 
-cos(a)*tan(a)/(1 + cot(a)^2)
Potencias [src] 
  / I*a    -I*a\                 
  |e      e    | /   -I*a    I*a\
I*|---- + -----|*\- e     + e   /
  \ 2       2  /                 
---------------------------------
   /       2   \ / I*a    -I*a\  
   \1 + cot (a)/*\e    + e    /
$$\frac{i \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right) \left(e^{i a} - e^{- i a}\right)}{\left(e^{i a} + e^{- i a}\right) \left(\cot^{2}{\left(a \right)} + 1\right)}$$ 
-cos(a)*tan(a) 
---------------
         2     
  1 + cot (a)
$$- \frac{\cos{\left(a \right)} \tan{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1}$$ 
-cos(a)*tan(a)/(1 + cot(a)^2)
Abrimos la expresión [src] 
-cos(a)*tan(a) 
---------------
         2     
  1 + cot (a)
$$- \frac{\cos{\left(a \right)} \tan{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1}$$ 
-cos(a)*tan(a)/(1 + cot(a)^2)
Denominador racional [src] 
-cos(a)*tan(a) 
---------------
         2     
  1 + cot (-a)
$$- \frac{\cos{\left(a \right)} \tan{\left(a \right)}}{\cot^{2}{\left(- a \right)} + 1}$$ 
-cos(a)*tan(a)/(1 + cot(-a)^2)
Parte trigonométrica [src] 
    3/    pi\
-cos |a - --|
     \    2 /
$$- \cos^{3}{\left(a - \frac{\pi}{2} \right)}$$ 
  -sin(a)  
-----------
       2   
    cos (a)
1 + -------
       2   
    sin (a)
$$- \frac{\sin{\left(a \right)}}{1 + \frac{\cos^{2}{\left(a \right)}}{\sin^{2}{\left(a \right)}}}$$ 
             -1               
------------------------------
/       2/    pi\\            
|    sec |a - --||            
|        \    2 /|    /    pi\
|1 + ------------|*sec|a - --|
|         2      |    \    2 /
\      sec (a)   /
$$- \frac{1}{\left(1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}\right) \sec{\left(a - \frac{\pi}{2} \right)}}$$ 
        3/a\  
  -8*tan |-|  
         \2/  
--------------
             3
/       2/a\\ 
|1 + tan |-|| 
\        \2//
$$- \frac{8 \tan^{3}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{3}}$$ 
  -1   
-------
   3   
csc (a)
$$- \frac{1}{\csc^{3}{\left(a \right)}}$$ 
    -1      
------------
   3/    pi\
sec |a - --|
    \    2 /
$$- \frac{1}{\sec^{3}{\left(a - \frac{\pi}{2} \right)}}$$ 
 /     /      pi\        /      pi\\ 
-|- cos|4*a - --| + 2*cos|2*a - --|| 
 \     \      2 /        \      2 // 
-------------------------------------
               8*cos(a)
$$- \frac{2 \cos{\left(2 a - \frac{\pi}{2} \right)} - \cos{\left(4 a - \frac{\pi}{2} \right)}}{8 \cos{\left(a \right)}}$$ 
-(-sin(4*a) + 2*sin(2*a)) 
--------------------------
         8*cos(a)
$$- \frac{2 \sin{\left(2 a \right)} - \sin{\left(4 a \right)}}{8 \cos{\left(a \right)}}$$ 
        3/a\  
  -8*cot |-|  
         \2/  
--------------
             3
/       2/a\\ 
|1 + cot |-|| 
\        \2//
$$- \frac{8 \cot^{3}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{3}}$$ 
-(-sin(4*a) + 2*sin(2*a)) 
--------------------------
           /    pi\       
      8*sin|a + --|       
           \    2 /
$$- \frac{2 \sin{\left(2 a \right)} - \sin{\left(4 a \right)}}{8 \sin{\left(a + \frac{\pi}{2} \right)}}$$ 
       2       /pi    \ 
 -2*sin (a)*sin|-- - a| 
               \2     / 
------------------------
/       2     \         
|    sin (2*a)|         
|1 + ---------|*sin(2*a)
|         4   |         
\    4*sin (a)/
$$- \frac{2 \sin^{2}{\left(a \right)} \sin{\left(- a + \frac{\pi}{2} \right)}}{\left(1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}\right) \sin{\left(2 a \right)}}$$ 
           -1            
-------------------------
/         2      \       
|      csc (a)   |       
|1 + ------------|*csc(a)
|       2/pi    \|       
|    csc |-- - a||       
\        \2     //
$$- \frac{1}{\left(\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(a \right)}}$$ 
 /       2/a\\ /    2*cot(2*a)      4*cot(a) \ 
-|1 + cot |-||*|- ------------- + -----------| 
 \        \2// |         2               2   | 
               \  1 + cot (2*a)   1 + cot (a)/ 
-----------------------------------------------
                  /        2/a\\               
                8*|-1 + cot |-||               
                  \         \2//
$$- \frac{\left(- \frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} + \frac{4 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1}\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}{8 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)}$$ 
    /       2/a\\          
   -|1 - tan |-||*tan(a)   
    \        \2//          
---------------------------
/       1   \ /       2/a\\
|1 + -------|*|1 + tan |-||
|       2   | \        \2//
\    tan (a)/
$$- \frac{\left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right) \tan{\left(a \right)}}{\left(1 + \frac{1}{\tan^{2}{\left(a \right)}}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$ 
    3   
-sin (a)
$$- \sin^{3}{\left(a \right)}$$ 
          /        2/a\\          
         -|-1 + cot |-||          
          \         \2//          
----------------------------------
/       2   \ /       2/a\\       
\1 + cot (a)/*|1 + cot |-||*cot(a)
              \        \2//
$$- \frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\cot^{2}{\left(a \right)} + 1\right) \cot{\left(a \right)}}$$ 
 /        1               2      \        
-|- ------------- + -------------|*sec(a) 
 |     /      pi\      /      pi\|        
 |  sec|4*a - --|   sec|2*a - --||        
 \     \      2 /      \      2 //        
------------------------------------------
                    8
$$- \frac{\left(- \frac{1}{\sec{\left(4 a - \frac{\pi}{2} \right)}} + \frac{2}{\sec{\left(2 a - \frac{\pi}{2} \right)}}\right) \sec{\left(a \right)}}{8}$$ 
         2              
   -2*sin (a)*cos(a)    
------------------------
/       2     \         
|    sin (2*a)|         
|1 + ---------|*sin(2*a)
|         4   |         
\    4*sin (a)/
$$- \frac{2 \sin^{2}{\left(a \right)} \cos{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}\right) \sin{\left(2 a \right)}}$$ 
      /     pi\  
   cos|-a - --|  
      \     2 /  
-----------------
          2      
       cos (a)   
1 + -------------
       2/     pi\
    cos |-a - --|
        \     2 /
$$\frac{\cos{\left(- a - \frac{\pi}{2} \right)}}{\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(- a - \frac{\pi}{2} \right)}} + 1}$$ 
 /       2/a\\ /    2*tan(2*a)      4*tan(a) \ 
-|1 + tan |-||*|- ------------- + -----------| 
 \        \2// |         2               2   | 
               \  1 + tan (2*a)   1 + tan (a)/ 
-----------------------------------------------
                  /       2/a\\                
                8*|1 - tan |-||                
                  \        \2//
$$- \frac{\left(- \frac{2 \tan{\left(2 a \right)}}{\tan^{2}{\left(2 a \right)} + 1} + \frac{4 \tan{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}{8 \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)}$$ 
        -1          
--------------------
/       2   \       
|    csc (a)|       
|1 + -------|*csc(a)
|       2   |       
\    sec (a)/
$$- \frac{1}{\left(\frac{\csc^{2}{\left(a \right)}}{\sec^{2}{\left(a \right)}} + 1\right) \csc{\left(a \right)}}$$ 
  3*sin(a)   sin(3*a)
- -------- + --------
     4          4
$$- \frac{3 \sin{\left(a \right)}}{4} + \frac{\sin{\left(3 a \right)}}{4}$$ 
 /     1          2    \    /pi    \ 
-|- -------- + --------|*csc|-- - a| 
 \  csc(4*a)   csc(2*a)/    \2     / 
-------------------------------------
                  8
$$- \frac{\left(- \frac{1}{\csc{\left(4 a \right)}} + \frac{2}{\csc{\left(2 a \right)}}\right) \csc{\left(- a + \frac{\pi}{2} \right)}}{8}$$ 
-(-1/csc(4*a) + 2/csc(2*a))*csc(pi/2 - a)/8
Unión de expresiones racionales [src] 
-cos(a)*tan(a) 
---------------
         2     
  1 + cot (a)
$$- \frac{\cos{\left(a \right)} \tan{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1}$$ 
-cos(a)*tan(a)/(1 + cot(a)^2)
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