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

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

Expresión a simplificar:

Solución

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