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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
   2                            
sin (a)    tan(2*a)       1     
-------- + -------- + ----------
cos(2*a)      2       tan(a) + 1
$$\left(\frac{\sin^{2}{\left(a \right)}}{\cos{\left(2 a \right)}} + \frac{\tan{\left(2 a \right)}}{2}\right) + \frac{1}{\tan{\left(a \right)} + 1}$$ 
sin(a)^2/cos(2*a) + tan(2*a)/2 + 1/(tan(a) + 1)
Simplificación general [src] 
                       4            
   1         2      sin (a)         
------- - sin (a) - ------- + tan(a)
   2                   2            
cos (a)             cos (a)         
------------------------------------
       (1 + tan(a))*cos(2*a)
$$\frac{- \frac{\sin^{4}{\left(a \right)}}{\cos^{2}{\left(a \right)}} - \sin^{2}{\left(a \right)} + \tan{\left(a \right)} + \frac{1}{\cos^{2}{\left(a \right)}}}{\left(\tan{\left(a \right)} + 1\right) \cos{\left(2 a \right)}}$$ 
(cos(a)^(-2) - sin(a)^2 - sin(a)^4/cos(a)^2 + tan(a))/((1 + tan(a))*cos(2*a))
Denominador común [src] 
              2         2                     
tan(2*a)   sin (a) + sin (a)*tan(a) + cos(2*a)
-------- + -----------------------------------
   2            cos(2*a)*tan(a) + cos(2*a)
$$\frac{\tan{\left(2 a \right)}}{2} + \frac{\sin^{2}{\left(a \right)} \tan{\left(a \right)} + \sin^{2}{\left(a \right)} + \cos{\left(2 a \right)}}{\cos{\left(2 a \right)} \tan{\left(a \right)} + \cos{\left(2 a \right)}}$$ 
tan(2*a)/2 + (sin(a)^2 + sin(a)^2*tan(a) + cos(2*a))/(cos(2*a)*tan(a) + cos(2*a))
Potencias [src] 
                           2    
    1        tan(2*a)   sin (a) 
---------- + -------- + --------
1 + tan(a)      2       cos(2*a)
$$\frac{\sin^{2}{\left(a \right)}}{\cos{\left(2 a \right)}} + \frac{\tan{\left(2 a \right)}}{2} + \frac{1}{\tan{\left(a \right)} + 1}$$ 
                                          2                           
                          /   -I*a    I*a\        /   2*I*a    -2*I*a\
          1               \- e     + e   /      I*\- e      + e      /
---------------------- - -------------------- + ----------------------
      /   I*a    -I*a\     / -2*I*a    2*I*a\      / -2*I*a    2*I*a\ 
    I*\- e    + e    /     |e         e     |    2*\e       + e     / 
1 + ------------------   4*|------- + ------|                         
        I*a    -I*a        \   2        2   /                         
       e    + e
$$- \frac{\left(e^{i a} - e^{- i a}\right)^{2}}{4 \left(\frac{e^{2 i a}}{2} + \frac{e^{- 2 i a}}{2}\right)} + \frac{i \left(- e^{2 i a} + e^{- 2 i a}\right)}{2 \left(e^{2 i a} + e^{- 2 i a}\right)} + \frac{1}{\frac{i \left(- e^{i a} + e^{- i a}\right)}{e^{i a} + e^{- i a}} + 1}$$ 
1/(1 + i*(-exp(i*a) + exp(-i*a))/(exp(i*a) + exp(-i*a))) - (-exp(-i*a) + exp(i*a))^2/(4*(exp(-2*i*a)/2 + exp(2*i*a)/2)) + i*(-exp(2*i*a) + exp(-2*i*a))/(2*(exp(-2*i*a) + exp(2*i*a)))
Combinatoria [src] 
     2                                            2                                     
2*sin (a) + 2*cos(2*a) + cos(2*a)*tan(2*a) + 2*sin (a)*tan(a) + cos(2*a)*tan(a)*tan(2*a)
----------------------------------------------------------------------------------------
                                2*(1 + tan(a))*cos(2*a)
$$\frac{2 \sin^{2}{\left(a \right)} \tan{\left(a \right)} + 2 \sin^{2}{\left(a \right)} + \cos{\left(2 a \right)} \tan{\left(a \right)} \tan{\left(2 a \right)} + \cos{\left(2 a \right)} \tan{\left(2 a \right)} + 2 \cos{\left(2 a \right)}}{2 \left(\tan{\left(a \right)} + 1\right) \cos{\left(2 a \right)}}$$ 
(2*sin(a)^2 + 2*cos(2*a) + cos(2*a)*tan(2*a) + 2*sin(a)^2*tan(a) + cos(2*a)*tan(a)*tan(2*a))/(2*(1 + tan(a))*cos(2*a))
Unión de expresiones racionales [src] 
                          /     2                       \
2*cos(2*a) + (1 + tan(a))*\2*sin (a) + cos(2*a)*tan(2*a)/
---------------------------------------------------------
                 2*(1 + tan(a))*cos(2*a)
$$\frac{\left(2 \sin^{2}{\left(a \right)} + \cos{\left(2 a \right)} \tan{\left(2 a \right)}\right) \left(\tan{\left(a \right)} + 1\right) + 2 \cos{\left(2 a \right)}}{2 \left(\tan{\left(a \right)} + 1\right) \cos{\left(2 a \right)}}$$ 
(2*cos(2*a) + (1 + tan(a))*(2*sin(a)^2 + cos(2*a)*tan(2*a)))/(2*(1 + tan(a))*cos(2*a))
Compilar la expresión [src] 
                           2    
    1        tan(2*a)   sin (a) 
---------- + -------- + --------
1 + tan(a)      2       cos(2*a)
$$\frac{\sin^{2}{\left(a \right)}}{\cos{\left(2 a \right)}} + \frac{\tan{\left(2 a \right)}}{2} + \frac{1}{\tan{\left(a \right)} + 1}$$ 
1/(1 + tan(a)) + tan(2*a)/2 + sin(a)^2/cos(2*a)
Abrimos la expresión [src] 
                                 2       
    1           tan(a)        sin (a)    
---------- + ----------- + --------------
tan(a) + 1          2                2   
             1 - tan (a)   -1 + 2*cos (a)
$$\frac{1}{\tan{\left(a \right)} + 1} + \frac{\sin^{2}{\left(a \right)}}{2 \cos^{2}{\left(a \right)} - 1} + \frac{\tan{\left(a \right)}}{1 - \tan^{2}{\left(a \right)}}$$ 
1/(tan(a) + 1) + tan(a)/(1 - tan(a)^2) + sin(a)^2/(-1 + 2*cos(a)^2)
Parte trigonométrica [src] 
                           /pi      \      /pi      \
                        csc|-- - 2*a|   csc|-- - 2*a|
  1          1             \2       /      \2       /
- - + --------------- + ------------- + -------------
  2          /pi    \         2           2*csc(2*a) 
          csc|-- - a|                                
             \2     /                                
      1 + -----------                                
             csc(a)
$$\frac{\csc{\left(- 2 a + \frac{\pi}{2} \right)}}{2} - \frac{1}{2} + \frac{\csc{\left(- 2 a + \frac{\pi}{2} \right)}}{2 \csc{\left(2 a \right)}} + \frac{1}{1 + \frac{\csc{\left(- a + \frac{\pi}{2} \right)}}{\csc{\left(a \right)}}}$$ 
                     1               
             -1 + --------           
    1             cos(2*a)   tan(2*a)
---------- + ------------- + --------
tan(a) + 1         2            2
$$\frac{-1 + \frac{1}{\cos{\left(2 a \right)}}}{2} + \frac{\tan{\left(2 a \right)}}{2} + \frac{1}{\tan{\left(a \right)} + 1}$$ 
                     1               
             -1 + --------           
    1             cos(2*a)   tan(2*a)
---------- + ------------- + --------
1 + tan(a)         2            2
$$\frac{-1 + \frac{1}{\cos{\left(2 a \right)}}}{2} + \frac{\tan{\left(2 a \right)}}{2} + \frac{1}{\tan{\left(a \right)} + 1}$$ 
                2                 
    1        sin (a)     sin(2*a) 
---------- + -------- + ----------
    sin(a)   cos(2*a)   2*cos(2*a)
1 + ------                        
    cos(a)
$$\frac{\sin^{2}{\left(a \right)}}{\cos{\left(2 a \right)}} + \frac{\sin{\left(2 a \right)}}{2 \cos{\left(2 a \right)}} + \frac{1}{\frac{\sin{\left(a \right)}}{\cos{\left(a \right)}} + 1}$$ 
                     /pi      \      /pi      \
                  csc|-- - 2*a|   csc|-- - 2*a|
       1             \2       /      \2       /
--------------- + ------------- + -------------
       /pi    \         2           2*csc(2*a) 
    csc|-- - a|      csc (a)                   
       \2     /                                
1 + -----------                                
       csc(a)
$$\frac{\csc{\left(- 2 a + \frac{\pi}{2} \right)}}{2 \csc{\left(2 a \right)}} + \frac{\csc{\left(- 2 a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(a \right)}} + \frac{1}{1 + \frac{\csc{\left(- a + \frac{\pi}{2} \right)}}{\csc{\left(a \right)}}}$$ 
                                       2     
  1       1        tan(2*a)     1 + tan (a)  
- - + ---------- + -------- + ---------------
  2   1 + tan(a)      2         /       2   \
                              2*\1 - tan (a)/
$$\frac{\tan{\left(2 a \right)}}{2} - \frac{1}{2} + \frac{1}{\tan{\left(a \right)} + 1} + \frac{\tan^{2}{\left(a \right)} + 1}{2 \left(1 - \tan^{2}{\left(a \right)}\right)}$$ 
    1        sec(2*a)    sec(2*a) 
---------- + -------- + ----------
    sec(a)      2       2*csc(2*a)
1 + ------   csc (a)              
    csc(a)
$$\frac{\sec{\left(2 a \right)}}{2 \csc{\left(2 a \right)}} + \frac{\sec{\left(2 a \right)}}{\csc^{2}{\left(a \right)}} + \frac{1}{1 + \frac{\sec{\left(a \right)}}{\csc{\left(a \right)}}}$$ 
                                         2      
  1       1            1          1 + cot (a)   
- - + ---------- + ---------- + ----------------
  2         1      2*cot(2*a)     /        2   \
      1 + ------                2*\-1 + cot (a)/
          cot(a)
$$- \frac{1}{2} + \frac{1}{2 \cot{\left(2 a \right)}} + \frac{\cot^{2}{\left(a \right)} + 1}{2 \left(\cot^{2}{\left(a \right)} - 1\right)} + \frac{1}{1 + \frac{1}{\cot{\left(a \right)}}}$$ 
  1       1            1        tan(2*a)
- - + ---------- + ---------- + --------
  2   1 + tan(a)   2*cos(2*a)      2
$$\frac{\tan{\left(2 a \right)}}{2} - \frac{1}{2} + \frac{1}{2 \cos{\left(2 a \right)}} + \frac{1}{\tan{\left(a \right)} + 1}$$ 
                                        /      pi\
                                     cos|2*a - --|
  1          1              1           \      2 /
- - + --------------- + ---------- + -------------
  2          /    pi\   2*cos(2*a)     2*cos(2*a) 
          cos|a - --|                             
             \    2 /                             
      1 + -----------                             
             cos(a)
$$- \frac{1}{2} + \frac{\cos{\left(2 a - \frac{\pi}{2} \right)}}{2 \cos{\left(2 a \right)}} + \frac{1}{2 \cos{\left(2 a \right)}} + \frac{1}{1 + \frac{\cos{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(a \right)}}}$$ 
                     2/    pi\      /      pi\
                  cos |a - --|   cos|2*a - --|
       1              \    2 /      \      2 /
--------------- + ------------ + -------------
       /    pi\     cos(2*a)       2*cos(2*a) 
    cos|a - --|                               
       \    2 /                               
1 + -----------                               
       cos(a)
$$\frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(2 a \right)}} + \frac{\cos{\left(2 a - \frac{\pi}{2} \right)}}{2 \cos{\left(2 a \right)}} + \frac{1}{1 + \frac{\cos{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(a \right)}}}$$ 
       1            sec(2*a)         sec(2*a)   
--------------- + ------------ + ---------------
       sec(a)        2/    pi\        /      pi\
1 + -----------   sec |a - --|   2*sec|2*a - --|
       /    pi\       \    2 /        \      2 /
    sec|a - --|                                 
       \    2 /
$$\frac{\sec{\left(2 a \right)}}{2 \sec{\left(2 a - \frac{\pi}{2} \right)}} + \frac{\sec{\left(2 a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + \frac{1}{\frac{\sec{\left(a \right)}}{\sec{\left(a - \frac{\pi}{2} \right)}} + 1}$$ 
  1       1        sec(2*a)   tan(2*a)
- - + ---------- + -------- + --------
  2   1 + tan(a)      2          2
$$\frac{\tan{\left(2 a \right)}}{2} + \frac{\sec{\left(2 a \right)}}{2} - \frac{1}{2} + \frac{1}{\tan{\left(a \right)} + 1}$$ 
                      2            2     
      1            sin (a)      sin (2*a)
------------- + ------------- + ---------
         2         /pi      \    sin(4*a)
    2*sin (a)   sin|-- + 2*a|            
1 + ---------      \2       /            
     sin(2*a)
$$\frac{\sin^{2}{\left(a \right)}}{\sin{\left(2 a + \frac{\pi}{2} \right)}} + \frac{\sin^{2}{\left(2 a \right)}}{\sin{\left(4 a \right)}} + \frac{1}{\frac{2 \sin^{2}{\left(a \right)}}{\sin{\left(2 a \right)}} + 1}$$ 
  1          1          sec(2*a)       sec(2*a)   
- - + --------------- + -------- + ---------------
  2          sec(a)        2            /      pi\
      1 + -----------              2*sec|2*a - --|
             /    pi\                   \      2 /
          sec|a - --|                             
             \    2 /
$$\frac{\sec{\left(2 a \right)}}{2} + \frac{\sec{\left(2 a \right)}}{2 \sec{\left(2 a - \frac{\pi}{2} \right)}} - \frac{1}{2} + \frac{1}{\frac{\sec{\left(a \right)}}{\sec{\left(a - \frac{\pi}{2} \right)}} + 1}$$ 
                                  2/a\ /       2   \   
                             4*cot |-|*\1 + cot (a)/   
    1            1                 \2/                 
---------- + ---------- + -----------------------------
      1      2*cot(2*a)                2               
1 + ------                /       2/a\\  /        2   \
    cot(a)                |1 + cot |-|| *\-1 + cot (a)/
                          \        \2//
$$\frac{1}{2 \cot{\left(2 a \right)}} + \frac{4 \left(\cot^{2}{\left(a \right)} + 1\right) \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2} \left(\cot^{2}{\left(a \right)} - 1\right)} + \frac{1}{1 + \frac{1}{\cot{\left(a \right)}}}$$ 
                   2          2     
      1         sin (a)    sin (2*a)
------------- + -------- + ---------
         2      cos(2*a)    sin(4*a)
    2*sin (a)                       
1 + ---------                       
     sin(2*a)
$$\frac{\sin^{2}{\left(a \right)}}{\cos{\left(2 a \right)}} + \frac{\sin^{2}{\left(2 a \right)}}{\sin{\left(4 a \right)}} + \frac{1}{\frac{2 \sin^{2}{\left(a \right)}}{\sin{\left(2 a \right)}} + 1}$$ 
                               2/a\ /       2   \   
                          4*tan |-|*\1 + tan (a)/   
    1        tan(2*a)           \2/                 
---------- + -------- + ----------------------------
1 + tan(a)      2                    2              
                        /       2/a\\  /       2   \
                        |1 + tan |-|| *\1 - tan (a)/
                        \        \2//
$$\frac{\tan{\left(2 a \right)}}{2} + \frac{1}{\tan{\left(a \right)} + 1} + \frac{4 \left(\tan^{2}{\left(a \right)} + 1\right) \tan^{2}{\left(\frac{a}{2} \right)}}{\left(1 - \tan^{2}{\left(a \right)}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$ 
                                           2     
  1         1                1          sin (2*a)
- - + ------------- + --------------- + ---------
  2            2           /pi      \    sin(4*a)
          2*sin (a)   2*sin|-- + 2*a|            
      1 + ---------        \2       /            
           sin(2*a)
$$\frac{\sin^{2}{\left(2 a \right)}}{\sin{\left(4 a \right)}} - \frac{1}{2} + \frac{1}{2 \sin{\left(2 a + \frac{\pi}{2} \right)}} + \frac{1}{\frac{2 \sin^{2}{\left(a \right)}}{\sin{\left(2 a \right)}} + 1}$$ 
-1/2 + 1/(1 + 2*sin(a)^2/sin(2*a)) + 1/(2*sin(pi/2 + 2*a)) + sin(2*a)^2/sin(4*a)
Denominador racional [src] 
                          /     2                       \
2*cos(2*a) + (1 + tan(a))*\2*sin (a) + cos(2*a)*tan(2*a)/
---------------------------------------------------------
                 2*(1 + tan(a))*cos(2*a)
$$\frac{\left(2 \sin^{2}{\left(a \right)} + \cos{\left(2 a \right)} \tan{\left(2 a \right)}\right) \left(\tan{\left(a \right)} + 1\right) + 2 \cos{\left(2 a \right)}}{2 \left(\tan{\left(a \right)} + 1\right) \cos{\left(2 a \right)}}$$ 
(2*cos(2*a) + (1 + tan(a))*(2*sin(a)^2 + cos(2*a)*tan(2*a)))/(2*(1 + tan(a))*cos(2*a))
Respuesta numérica [src] 
1/(1.0 + tan(a)) + 0.5*tan(2*a) + sin(a)^2/cos(2*a)
1/(1.0 + tan(a)) + 0.5*tan(2*a) + sin(a)^2/cos(2*a)
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