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Simplificación de expresiones/ Descomposición en fracciones simples/ cos(2*a)*tan(2*a)/((2*sin(a)))

¿Cómo vas a descomponer esta cos(2*a)*tan(2*a)/((2*sin(a))) expresión en fracciones?

Expresión a simplificar:

Solución

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