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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)
cos⁡(a)tan⁡(a)cot⁡2(a)+1\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(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)
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    /
i(−eia+e−ia)(eia2+e−ia2)(eia+e−ia)(cot⁡2(a)+1)\frac{i \left(- e^{i a} + e^{- i a}\right) \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)}{\left(e^{i a} + e^{- i a}\right) \left(\cot^{2}{\left(a \right)} + 1\right)} 
i*(exp(i*a)/2 + exp(-i*a)/2)*(-exp(i*a) + exp(-i*a))/((1 + cot(a)^2)*(exp(i*a) + exp(-i*a)))
Parte trigonométrica [src] 
         2              
    2*sin (a)*cos(a)    
------------------------
/       2     \         
|    sin (2*a)|         
|1 + ---------|*sin(2*a)
|         4   |         
\    4*sin (a)/
2sin⁡2(a)cos⁡(a)(1+sin⁡2(2a)4sin⁡4(a))sin⁡(2a)\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)}} 
-sin(4*a) + 2*sin(2*a)
----------------------
         /    pi\     
    8*sin|a + --|     
         \    2 /
2sin⁡(2a)−sin⁡(4a)8sin⁡(a+π2)\frac{2 \sin{\left(2 a \right)} - \sin{\left(4 a \right)}}{8 \sin{\left(a + \frac{\pi}{2} \right)}} 
     /    pi\   
  cos|a - --|   
     \    2 /   
----------------
         2      
      cos (a)   
1 + ------------
       2/    pi\
    cos |a - --|
        \    2 /
cos⁡(a−π2)cos⁡2(a)cos⁡2(a−π2)+1\frac{\cos{\left(a - \frac{\pi}{2} \right)}}{\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} + 1} 
  sin(3*a)   3*sin(a)
- -------- + --------
     4          4
3sin⁡(a)4−sin⁡(3a)4\frac{3 \sin{\left(a \right)}}{4} - \frac{\sin{\left(3 a \right)}}{4} 
     /      pi\        /      pi\
- cos|4*a - --| + 2*cos|2*a - --|
     \      2 /        \      2 /
---------------------------------
             8*cos(a)
2cos⁡(2a−π2)−cos⁡(4a−π2)8cos⁡(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)
2sin⁡(2a)−sin⁡(4a)8cos⁡(a)\frac{2 \sin{\left(2 a \right)} - \sin{\left(4 a \right)}}{8 \cos{\left(a \right)}} 
   1   
-------
   3   
csc (a)
1csc⁡3(a)\frac{1}{\csc^{3}{\left(a \right)}} 
              1               
------------------------------
/       2/    pi\\            
|    sec |a - --||            
|        \    2 /|    /    pi\
|1 + ------------|*sec|a - --|
|         2      |    \    2 /
\      sec (a)   /
1(1+sec⁡2(a−π2)sec⁡2(a))sec⁡(a−π2)\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)}} 
   sin(a)  
-----------
       2   
    cos (a)
1 + -------
       2   
    sin (a)
sin⁡(a)1+cos⁡2(a)sin⁡2(a)\frac{\sin{\left(a \right)}}{1 + \frac{\cos^{2}{\left(a \right)}}{\sin^{2}{\left(a \right)}}} 
    /       2/a\\          
    |1 - tan |-||*tan(a)   
    \        \2//          
---------------------------
/       1   \ /       2/a\\
|1 + -------|*|1 + tan |-||
|       2   | \        \2//
\    tan (a)/
(1−tan⁡2(a2))tan⁡(a)(1+1tan⁡2(a))(tan⁡2(a2)+1)\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)} 
/       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//
(−2tan⁡(2a)tan⁡2(2a)+1+4tan⁡(a)tan⁡2(a)+1)(tan⁡2(a2)+1)8(1−tan⁡2(a2))\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)} 
       3/a\   
  8*cot |-|   
        \2/   
--------------
             3
/       2/a\\ 
|1 + cot |-|| 
\        \2//
8cot⁡3(a2)(cot⁡2(a2)+1)3\frac{8 \cot^{3}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{3}} 
   3   
sin (a)
sin⁡3(a)\sin^{3}{\left(a \right)} 
   3/    pi\
cos |a - --|
    \    2 /
cos⁡3(a−π2)\cos^{3}{\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)/
2sin⁡2(a)sin⁡(a+π2)(1+sin⁡2(2a)4sin⁡4(a))sin⁡(2a)\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      \       
|- ------------- + -------------|*sec(a)
|     /      pi\      /      pi\|       
|  sec|4*a - --|   sec|2*a - --||       
\     \      2 /      \      2 //       
----------------------------------------
                   8
(−1sec⁡(4a−π2)+2sec⁡(2a−π2))sec⁡(a)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} 
     1      
------------
   3/    pi\
sec |a - --|
    \    2 /
1sec⁡3(a−π2)\frac{1}{\sec^{3}{\left(a - \frac{\pi}{2} \right)}} 
         1          
--------------------
/       2   \       
|    csc (a)|       
|1 + -------|*csc(a)
|       2   |       
\    sec (a)/
1(csc⁡2(a)sec⁡2(a)+1)csc⁡(a)\frac{1}{\left(\frac{\csc^{2}{\left(a \right)}}{\sec^{2}{\left(a \right)}} + 1\right) \csc{\left(a \right)}} 
            1            
-------------------------
/         2      \       
|      csc (a)   |       
|1 + ------------|*csc(a)
|       2/pi    \|       
|    csc |-- - a||       
\        \2     //
1(csc⁡2(a)csc⁡2(−a+π2)+1)csc⁡(a)\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\           
           -1 + cot |-|           
                    \2/           
----------------------------------
/       2   \ /       2/a\\       
\1 + cot (a)/*|1 + cot |-||*cot(a)
              \        \2//
cot⁡2(a2)−1(cot⁡2(a2)+1)(cot⁡2(a)+1)cot⁡(a)\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    \    /pi    \
|- -------- + --------|*csc|-- - a|
\  csc(4*a)   csc(2*a)/    \2     /
-----------------------------------
                 8
(−1csc⁡(4a)+2csc⁡(2a))csc⁡(−a+π2)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} 
       3/a\   
  8*tan |-|   
        \2/   
--------------
             3
/       2/a\\ 
|1 + tan |-|| 
\        \2//
8tan⁡3(a2)(tan⁡2(a2)+1)3\frac{8 \tan^{3}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{3}} 
/       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//
(−2cot⁡(2a)cot⁡2(2a)+1+4cot⁡(a)cot⁡2(a)+1)(cot⁡2(a2)+1)8(cot⁡2(a2)−1)\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)} 
(1 + cot(a/2)^2)*(-2*cot(2*a)/(1 + cot(2*a)^2) + 4*cot(a)/(1 + cot(a)^2))/(8*(-1 + cot(a/2)^2))
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