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Simplificación de expresiones/ Descomposición en fracciones simples/ sin(a)^(2)/(1+tan(a)^(2))+cos(a)^(2)/(1+cot(a)^(2))

¿Cómo vas a descomponer esta sin(a)^(2)/(1+tan(a)^(2))+cos(a)^(2)/(1+cot(a)^(2)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
     2             2     
  sin (a)       cos (a)  
----------- + -----------
       2             2   
1 + tan (a)   1 + cot (a)
$$\frac{\cos^{2}{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} + \frac{\sin^{2}{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1}$$ 
sin(a)^2/(1 + tan(a)^2) + cos(a)^2/(1 + cot(a)^2)
Simplificación general [src] 
1   cos(4*a)
- - --------
4      4
$$\frac{1}{4} - \frac{\cos{\left(4 a \right)}}{4}$$ 
1/4 - cos(4*a)/4
Respuesta numérica [src] 
cos(a)^2/(1.0 + cot(a)^2) + sin(a)^2/(1.0 + tan(a)^2)
cos(a)^2/(1.0 + cot(a)^2) + sin(a)^2/(1.0 + tan(a)^2)
Denominador racional [src] 
   2    /       2   \      2    /       2   \
cos (a)*\1 + tan (a)/ + sin (a)*\1 + cot (a)/
---------------------------------------------
         /       2   \ /       2   \         
         \1 + cot (a)/*\1 + tan (a)/
$$\frac{\left(\tan^{2}{\left(a \right)} + 1\right) \cos^{2}{\left(a \right)} + \left(\cot^{2}{\left(a \right)} + 1\right) \sin^{2}{\left(a \right)}}{\left(\tan^{2}{\left(a \right)} + 1\right) \left(\cot^{2}{\left(a \right)} + 1\right)}$$ 
(cos(a)^2*(1 + tan(a)^2) + sin(a)^2*(1 + cot(a)^2))/((1 + cot(a)^2)*(1 + tan(a)^2))
Combinatoria [src] 
   2         2         2       2         2       2   
cos (a) + sin (a) + cos (a)*tan (a) + cot (a)*sin (a)
-----------------------------------------------------
             /       2   \ /       2   \             
             \1 + cot (a)/*\1 + tan (a)/
$$\frac{\sin^{2}{\left(a \right)} \cot^{2}{\left(a \right)} + \sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)} \tan^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}}{\left(\tan^{2}{\left(a \right)} + 1\right) \left(\cot^{2}{\left(a \right)} + 1\right)}$$ 
(cos(a)^2 + sin(a)^2 + cos(a)^2*tan(a)^2 + cot(a)^2*sin(a)^2)/((1 + cot(a)^2)*(1 + tan(a)^2))
Potencias [src] 
              2                            
/ I*a    -I*a\                             
|e      e    |                        2    
|---- + -----|        /   -I*a    I*a\     
\ 2       2  /        \- e     + e   /     
--------------- - -------------------------
         2          /                    2\
  1 + cot (a)       |    /   I*a    -I*a\ |
                    |    \- e    + e    / |
                  4*|1 - -----------------|
                    |                   2 |
                    |     / I*a    -I*a\  |
                    \     \e    + e    /  /
$$\frac{\left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)^{2}}{\cot^{2}{\left(a \right)} + 1} - \frac{\left(e^{i a} - e^{- i a}\right)^{2}}{4 \left(- \frac{\left(- e^{i a} + e^{- i a}\right)^{2}}{\left(e^{i a} + e^{- i a}\right)^{2}} + 1\right)}$$ 
(exp(i*a)/2 + exp(-i*a)/2)^2/(1 + cot(a)^2) - (-exp(-i*a) + exp(i*a))^2/(4*(1 - (-exp(i*a) + exp(-i*a))^2/(exp(i*a) + exp(-i*a))^2))
Unión de expresiones racionales [src] 
   2    /       2   \      2    /       2   \
cos (a)*\1 + tan (a)/ + sin (a)*\1 + cot (a)/
---------------------------------------------
         /       2   \ /       2   \         
         \1 + cot (a)/*\1 + tan (a)/
$$\frac{\left(\tan^{2}{\left(a \right)} + 1\right) \cos^{2}{\left(a \right)} + \left(\cot^{2}{\left(a \right)} + 1\right) \sin^{2}{\left(a \right)}}{\left(\tan^{2}{\left(a \right)} + 1\right) \left(\cot^{2}{\left(a \right)} + 1\right)}$$ 
(cos(a)^2*(1 + tan(a)^2) + sin(a)^2*(1 + cot(a)^2))/((1 + cot(a)^2)*(1 + tan(a)^2))
Denominador común [src] 
   2         2         2       2         2       2   
cos (a) + sin (a) + cos (a)*tan (a) + cot (a)*sin (a)
-----------------------------------------------------
              2         2         2       2          
       1 + cot (a) + tan (a) + cot (a)*tan (a)
$$\frac{\sin^{2}{\left(a \right)} \cot^{2}{\left(a \right)} + \sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)} \tan^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}}{\tan^{2}{\left(a \right)} \cot^{2}{\left(a \right)} + \tan^{2}{\left(a \right)} + \cot^{2}{\left(a \right)} + 1}$$ 
(cos(a)^2 + sin(a)^2 + cos(a)^2*tan(a)^2 + cot(a)^2*sin(a)^2)/(1 + cot(a)^2 + tan(a)^2 + cot(a)^2*tan(a)^2)
Parte trigonométrica [src] 
            1                               1               
-------------------------- + -------------------------------
/       2/pi    \\           /         2      \             
|    csc |-- - a||           |      csc (a)   |    2/pi    \
|        \2     /|    2      |1 + ------------|*csc |-- - a|
|1 + ------------|*csc (a)   |       2/pi    \|     \2     /
|         2      |           |    csc |-- - a||             
\      csc (a)   /           \        \2     //
$$\frac{1}{\left(\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} + 1\right) \csc^{2}{\left(- a + \frac{\pi}{2} \right)}} + \frac{1}{\left(1 + \frac{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(a \right)}}\right) \csc^{2}{\left(a \right)}}$$ 
   2    /       2           2/a\        4/a\\
sin (a)*|2 - sin (a) - 4*sin |-| + 4*sin |-||
        \                    \2/         \2//
$$\left(4 \sin^{4}{\left(\frac{a}{2} \right)} - 4 \sin^{2}{\left(\frac{a}{2} \right)} - \sin^{2}{\left(a \right)} + 2\right) \sin^{2}{\left(a \right)}$$ 
     2/    pi\                     
  cos |a - --|            2        
      \    2 /         cos (a)     
---------------- + ----------------
       2/    pi\            2      
    cos |a - --|         cos (a)   
        \    2 /   1 + ------------
1 + ------------          2/    pi\
         2             cos |a - --|
      cos (a)              \    2 /
$$\frac{\cos^{2}{\left(a \right)}}{\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} + 1} + \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{1 + \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(a \right)}}}$$ 
          /            2/a\            2/a\              4/a\  \
          |      16*cot |-|       4*cot |-|        64*cot |-|  |
     2/a\ |             \4/             \2/               \4/  |
4*cot |-|*|2 - -------------- - -------------- + --------------|
      \2/ |                 2                2                4|
          |    /       2/a\\    /       2/a\\    /       2/a\\ |
          |    |1 + cot |-||    |1 + cot |-||    |1 + cot |-|| |
          \    \        \4//    \        \2//    \        \4// /
----------------------------------------------------------------
                                      2                         
                         /       2/a\\                          
                         |1 + cot |-||                          
                         \        \2//
$$\frac{4 \left(2 - \frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} - \frac{16 \cot^{2}{\left(\frac{a}{4} \right)}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}} + \frac{64 \cot^{4}{\left(\frac{a}{4} \right)}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{4}}\right) \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$ 
                    2                                      
      /        2/a\\                         2/a\          
      |-1 + cot |-||                    4*cot |-|          
      \         \2//                          \2/          
---------------------------- + ----------------------------
                           2                              2
/       2   \ /       2/a\\    /       1   \ /       2/a\\ 
\1 + cot (a)/*|1 + cot |-||    |1 + -------|*|1 + cot |-|| 
              \        \2//    |       2   | \        \2// 
                               \    cot (a)/
$$\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2} \left(\cot^{2}{\left(a \right)} + 1\right)} + \frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(1 + \frac{1}{\cot^{2}{\left(a \right)}}\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$ 
              2      
1     -1 + cot (2*a) 
- - -----------------
4     /       2     \
    4*\1 + cot (2*a)/
$$- \frac{\cot^{2}{\left(2 a \right)} - 1}{4 \left(\cot^{2}{\left(2 a \right)} + 1\right)} + \frac{1}{4}$$ 
   2/    pi\ /       2/    pi\        2/a   pi\        4/a   pi\\
cos |a - --|*|2 - cos |a - --| - 4*cos |- - --| + 4*cos |- - --||
    \    2 / \        \    2 /         \2   2 /         \2   2 //
$$\left(4 \cos^{4}{\left(\frac{a}{2} - \frac{\pi}{2} \right)} - 4 \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)} - \cos^{2}{\left(a - \frac{\pi}{2} \right)} + 2\right) \cos^{2}{\left(a - \frac{\pi}{2} \right)}$$ 
     4/a\    2/a\ /                     1 - cos(2*a)         8/a\    4/a\\
4*cos |-|*tan |-|*|2 - 2*(1 - cos(a)) - ------------ + 64*cos |-|*tan |-||
      \2/     \2/ \                          2                \4/     \4//
$$4 \left(- 2 \left(1 - \cos{\left(a \right)}\right) - \frac{1 - \cos{\left(2 a \right)}}{2} + 64 \cos^{8}{\left(\frac{a}{4} \right)} \tan^{4}{\left(\frac{a}{4} \right)} + 2\right) \cos^{4}{\left(\frac{a}{2} \right)} \tan^{2}{\left(\frac{a}{2} \right)}$$ 
         1              4              4      
2 - ------------ - ------------ + ------------
       2/    pi\      2/a   pi\      4/a   pi\
    sec |a - --|   sec |- - --|   sec |- - --|
        \    2 /       \2   2 /       \2   2 /
----------------------------------------------
                    2/    pi\                 
                 sec |a - --|                 
                     \    2 /
$$\frac{2 - \frac{1}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} - \frac{4}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + \frac{4}{\sec^{4}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}$$ 
          /            2/a\            2/a\              4/a\  \
          |      16*tan |-|       4*tan |-|        64*tan |-|  |
     2/a\ |             \4/             \2/               \4/  |
4*tan |-|*|2 - -------------- - -------------- + --------------|
      \2/ |                 2                2                4|
          |    /       2/a\\    /       2/a\\    /       2/a\\ |
          |    |1 + tan |-||    |1 + tan |-||    |1 + tan |-|| |
          \    \        \4//    \        \2//    \        \4// /
----------------------------------------------------------------
                                      2                         
                         /       2/a\\                          
                         |1 + tan |-||                          
                         \        \2//
$$\frac{4 \left(2 - \frac{4 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} - \frac{16 \tan^{2}{\left(\frac{a}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}} + \frac{64 \tan^{4}{\left(\frac{a}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{4}}\right) \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$ 
                    2                                      
       /       2/a\\                         2/a\          
       |1 - tan |-||                    4*tan |-|          
       \        \2//                          \2/          
---------------------------- + ----------------------------
                           2                              2
/       1   \ /       2/a\\    /       2   \ /       2/a\\ 
|1 + -------|*|1 + tan |-||    \1 + tan (a)/*|1 + tan |-|| 
|       2   | \        \2//                  \        \2// 
\    tan (a)/
$$\frac{\left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(a \right)}}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} + \frac{4 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2} \left(\tan^{2}{\left(a \right)} + 1\right)}$$ 
      2               2      
   sin (a)         cos (a)   
------------- + -------------
         4             2     
    4*sin (a)       sin (2*a)
1 + ---------   1 + ---------
       2                 4   
    sin (2*a)       4*sin (a)
$$\frac{\sin^{2}{\left(a \right)}}{\frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\left(2 a \right)}} + 1} + \frac{\cos^{2}{\left(a \right)}}{1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}}$$ 
          1                       1          
--------------------- + ---------------------
/       2   \           /       2   \        
|    sec (a)|    2      |    csc (a)|    2   
|1 + -------|*csc (a)   |1 + -------|*sec (a)
|       2   |           |       2   |        
\    csc (a)/           \    sec (a)/
$$\frac{1}{\left(\frac{\csc^{2}{\left(a \right)}}{\sec^{2}{\left(a \right)}} + 1\right) \sec^{2}{\left(a \right)}} + \frac{1}{\left(1 + \frac{\sec^{2}{\left(a \right)}}{\csc^{2}{\left(a \right)}}\right) \csc^{2}{\left(a \right)}}$$ 
     2/a\    4/a\ /                     1 - cos(2*a)         4/a\    8/a\\
4*cot |-|*sin |-|*|2 - 2*(1 - cos(a)) - ------------ + 64*cot |-|*sin |-||
      \2/     \2/ \                          2                \4/     \4//
$$4 \left(- 2 \left(1 - \cos{\left(a \right)}\right) - \frac{1 - \cos{\left(2 a \right)}}{2} + 64 \sin^{8}{\left(\frac{a}{4} \right)} \cot^{4}{\left(\frac{a}{4} \right)} + 2\right) \sin^{4}{\left(\frac{a}{2} \right)} \cot^{2}{\left(\frac{a}{2} \right)}$$ 
1   cos(4*a)
- - --------
4      4
$$\frac{1}{4} - \frac{\cos{\left(4 a \right)}}{4}$$ 
1          1       
- - ---------------
4        /pi      \
    4*csc|-- - 4*a|
         \2       /
$$\frac{1}{4} - \frac{1}{4 \csc{\left(- 4 a + \frac{\pi}{2} \right)}}$$ 
1       1     
- - ----------
4   4*sec(4*a)
$$\frac{1}{4} - \frac{1}{4 \sec{\left(4 a \right)}}$$ 
       /pi      \
    sin|-- + 4*a|
1      \2       /
- - -------------
4         4
$$\frac{1}{4} - \frac{\sin{\left(4 a + \frac{\pi}{2} \right)}}{4}$$ 
                    2/    pi\
      2          sin |a + --|
   sin (a)           \    2 /
------------- + -------------
         4             2     
    4*sin (a)       sin (2*a)
1 + ---------   1 + ---------
       2                 4   
    sin (2*a)       4*sin (a)
$$\frac{\sin^{2}{\left(a \right)}}{\frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\left(2 a \right)}} + 1} + \frac{\sin^{2}{\left(a + \frac{\pi}{2} \right)}}{1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}}$$ 
            1                               1               
-------------------------- + -------------------------------
/       2/    pi\\           /         2      \             
|    sec |a - --||           |      sec (a)   |    2/    pi\
|        \    2 /|    2      |1 + ------------|*sec |a - --|
|1 + ------------|*sec (a)   |       2/    pi\|     \    2 /
|         2      |           |    sec |a - --||             
\      sec (a)   /           \        \    2 //
$$\frac{1}{\left(\frac{\sec^{2}{\left(a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + 1\right) \sec^{2}{\left(a - \frac{\pi}{2} \right)}} + \frac{1}{\left(1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}\right) \sec^{2}{\left(a \right)}}$$ 
             2       
1     1 - tan (2*a)  
- - -----------------
4     /       2     \
    4*\1 + tan (2*a)/
$$- \frac{1 - \tan^{2}{\left(2 a \right)}}{4 \left(\tan^{2}{\left(2 a \right)} + 1\right)} + \frac{1}{4}$$ 
     2             2     
  sin (a)       cos (a)  
----------- + -----------
       2             2   
    sin (a)       cos (a)
1 + -------   1 + -------
       2             2   
    cos (a)       sin (a)
$$\frac{\sin^{2}{\left(a \right)}}{\frac{\sin^{2}{\left(a \right)}}{\cos^{2}{\left(a \right)}} + 1} + \frac{\cos^{2}{\left(a \right)}}{1 + \frac{\cos^{2}{\left(a \right)}}{\sin^{2}{\left(a \right)}}}$$ 
       1         4         4   
2 - ------- - ------- + -------
       2         2/a\      4/a\
    csc (a)   csc |-|   csc |-|
                  \2/       \2/
-------------------------------
               2               
            csc (a)
$$\frac{2 - \frac{1}{\csc^{2}{\left(a \right)}} - \frac{4}{\csc^{2}{\left(\frac{a}{2} \right)}} + \frac{4}{\csc^{4}{\left(\frac{a}{2} \right)}}}{\csc^{2}{\left(a \right)}}$$ 
(2 - 1/csc(a)^2 - 4/csc(a/2)^2 + 4/csc(a/2)^4)/csc(a)^2
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