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

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

Expresión a simplificar:

Solución

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