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

¿Cómo vas a descomponer esta sin(a)^2/cos(a)^2+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     
------- + -------- + ----------
   2         2       tan(a) + 1
cos (a)
(sin⁡2(a)cos⁡2(a)+tan⁡(2a)2)+1tan⁡(a)+1\left(\frac{\sin^{2}{\left(a \right)}}{\cos^{2}{\left(a \right)}} + \frac{\tan{\left(2 a \right)}}{2}\right) + \frac{1}{\tan{\left(a \right)} + 1} 
sin(a)^2/cos(a)^2 + tan(2*a)/2 + 1/(tan(a) + 1)
Simplificación general [src] 
                           2   
    1        tan(2*a)   sin (a)
---------- + -------- + -------
1 + tan(a)      2          2   
                        cos (a)
sin⁡2(a)cos⁡2(a)+tan⁡(2a)2+1tan⁡(a)+1\frac{\sin^{2}{\left(a \right)}}{\cos^{2}{\left(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(a)^2
Respuesta numérica [src] 
1/(1.0 + tan(a)) + 0.5*tan(2*a) + sin(a)^2/cos(a)^2
1/(1.0 + tan(a)) + 0.5*tan(2*a) + sin(a)^2/cos(a)^2
Denominador racional [src] 
     2                   /     2         2            \
2*cos (a) + (1 + tan(a))*\2*sin (a) + cos (a)*tan(2*a)/
-------------------------------------------------------
                                   2                   
                 2*(1 + tan(a))*cos (a)
(2sin⁡2(a)+cos⁡2(a)tan⁡(2a))(tan⁡(a)+1)+2cos⁡2(a)2(tan⁡(a)+1)cos⁡2(a)\frac{\left(2 \sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)} \tan{\left(2 a \right)}\right) \left(\tan{\left(a \right)} + 1\right) + 2 \cos^{2}{\left(a \right)}}{2 \left(\tan{\left(a \right)} + 1\right) \cos^{2}{\left(a \right)}} 
(2*cos(a)^2 + (1 + tan(a))*(2*sin(a)^2 + cos(a)^2*tan(2*a)))/(2*(1 + tan(a))*cos(a)^2)
Potencias [src] 
                                         2                         
                         /   -I*a    I*a\      /   2*I*a    -2*I*a\
          1              \- e     + e   /    I*\- e      + e      /
---------------------- - ----------------- + ----------------------
      /   I*a    -I*a\                   2      / -2*I*a    2*I*a\ 
    I*\- e    + e    /     / I*a    -I*a\     2*\e       + e     / 
1 + ------------------     |e      e    |                          
        I*a    -I*a      4*|---- + -----|                          
       e    + e            \ 2       2  /
i(−e2ia+e−2ia)2(e2ia+e−2ia)−(eia−e−ia)24(eia2+e−ia2)2+1i(−eia+e−ia)eia+e−ia+1\frac{i \left(- e^{2 i a} + e^{- 2 i a}\right)}{2 \left(e^{2 i a} + e^{- 2 i a}\right)} - \frac{\left(e^{i a} - e^{- i a}\right)^{2}}{4 \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)^{2}} + \frac{1}{\frac{i \left(- e^{i a} + e^{- i a}\right)}{e^{i a} + e^{- i a}} + 1} 
                           2   
    1        tan(2*a)   sin (a)
---------- + -------- + -------
1 + tan(a)      2          2   
                        cos (a)
sin⁡2(a)cos⁡2(a)+tan⁡(2a)2+1tan⁡(a)+1\frac{\sin^{2}{\left(a \right)}}{\cos^{2}{\left(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(a)^2
Denominador común [src] 
              2         2         2          
tan(2*a)   cos (a) + sin (a) + sin (a)*tan(a)
-------- + ----------------------------------
   2               2         2               
                cos (a) + cos (a)*tan(a)
tan⁡(2a)2+sin⁡2(a)tan⁡(a)+sin⁡2(a)+cos⁡2(a)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^{2}{\left(a \right)}}{\cos^{2}{\left(a \right)} \tan{\left(a \right)} + \cos^{2}{\left(a \right)}} 
tan(2*a)/2 + (cos(a)^2 + sin(a)^2 + sin(a)^2*tan(a))/(cos(a)^2 + cos(a)^2*tan(a))
Unión de expresiones racionales [src] 
     2                   /     2         2            \
2*cos (a) + (1 + tan(a))*\2*sin (a) + cos (a)*tan(2*a)/
-------------------------------------------------------
                                   2                   
                 2*(1 + tan(a))*cos (a)
(2sin⁡2(a)+cos⁡2(a)tan⁡(2a))(tan⁡(a)+1)+2cos⁡2(a)2(tan⁡(a)+1)cos⁡2(a)\frac{\left(2 \sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)} \tan{\left(2 a \right)}\right) \left(\tan{\left(a \right)} + 1\right) + 2 \cos^{2}{\left(a \right)}}{2 \left(\tan{\left(a \right)} + 1\right) \cos^{2}{\left(a \right)}} 
(2*cos(a)^2 + (1 + tan(a))*(2*sin(a)^2 + cos(a)^2*tan(2*a)))/(2*(1 + tan(a))*cos(a)^2)
Combinatoria [src] 
     2           2         2                    2                2                   
2*cos (a) + 2*sin (a) + cos (a)*tan(2*a) + 2*sin (a)*tan(a) + cos (a)*tan(a)*tan(2*a)
-------------------------------------------------------------------------------------
                                                  2                                  
                                2*(1 + tan(a))*cos (a)
2sin⁡2(a)tan⁡(a)+2sin⁡2(a)+cos⁡2(a)tan⁡(a)tan⁡(2a)+cos⁡2(a)tan⁡(2a)+2cos⁡2(a)2(tan⁡(a)+1)cos⁡2(a)\frac{2 \sin^{2}{\left(a \right)} \tan{\left(a \right)} + 2 \sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)} \tan{\left(a \right)} \tan{\left(2 a \right)} + \cos^{2}{\left(a \right)} \tan{\left(2 a \right)} + 2 \cos^{2}{\left(a \right)}}{2 \left(\tan{\left(a \right)} + 1\right) \cos^{2}{\left(a \right)}} 
(2*cos(a)^2 + 2*sin(a)^2 + cos(a)^2*tan(2*a) + 2*sin(a)^2*tan(a) + cos(a)^2*tan(a)*tan(2*a))/(2*(1 + tan(a))*cos(a)^2)
Abrimos la expresión [src] 
                              2   
    1           tan(a)     sin (a)
---------- + ----------- + -------
tan(a) + 1          2         2   
             1 - tan (a)   cos (a)
sin⁡2(a)cos⁡2(a)+1tan⁡(a)+1+tan⁡(a)1−tan⁡2(a)\frac{\sin^{2}{\left(a \right)}}{\cos^{2}{\left(a \right)}} + \frac{1}{\tan{\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/cos(a)^2
Compilar la expresión [src] 
                           2   
    1        tan(2*a)   sin (a)
---------- + -------- + -------
1 + tan(a)      2          2   
                        cos (a)
sin⁡2(a)cos⁡2(a)+tan⁡(2a)2+1tan⁡(a)+1\frac{\sin^{2}{\left(a \right)}}{\cos^{2}{\left(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(a)^2
Parte trigonométrica [src] 
                       2                        
       1            sec (a)          sec(2*a)   
--------------- + ------------ + ---------------
       sec(a)        2/    pi\        /      pi\
1 + -----------   sec |a - --|   2*sec|2*a - --|
       /    pi\       \    2 /        \      2 /
    sec|a - --|                                 
       \    2 /
sec⁡2(a)sec⁡2(a−π2)+sec⁡(2a)2sec⁡(2a−π2)+1sec⁡(a)sec⁡(a−π2)+1\frac{\sec^{2}{\left(a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + \frac{\sec{\left(2 a \right)}}{2 \sec{\left(2 a - \frac{\pi}{2} \right)}} + \frac{1}{\frac{\sec{\left(a \right)}}{\sec{\left(a - \frac{\pi}{2} \right)}} + 1} 
                     2/    pi\      /      pi\
                  cos |a - --|   cos|2*a - --|
       1              \    2 /      \      2 /
--------------- + ------------ + -------------
       /    pi\        2           2*cos(2*a) 
    cos|a - --|     cos (a)                   
       \    2 /                               
1 + -----------                               
       cos(a)
cos⁡(2a−π2)2cos⁡(2a)+cos⁡2(a−π2)cos⁡2(a)+11+cos⁡(a−π2)cos⁡(a)\frac{\cos{\left(2 a - \frac{\pi}{2} \right)}}{2 \cos{\left(2 a \right)}} + \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(a \right)}} + \frac{1}{1 + \frac{\cos{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(a \right)}}} 
                                  2/a\   
                             4*cot |-|   
    1            1                 \2/   
---------- + ---------- + ---------------
      1      2*cot(2*a)                 2
1 + ------                /        2/a\\ 
    cot(a)                |-1 + cot |-|| 
                          \         \2//
12cot⁡(2a)+4cot⁡2(a2)(cot⁡2(a2)−1)2+11+1cot⁡(a)\frac{1}{2 \cot{\left(2 a \right)}} + \frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}} + \frac{1}{1 + \frac{1}{\cot{\left(a \right)}}} 
    1           2      tan(2*a)
---------- + tan (a) + --------
1 + tan(a)                2
tan⁡2(a)+tan⁡(2a)2+1tan⁡(a)+1\tan^{2}{\left(a \right)} + \frac{\tan{\left(2 a \right)}}{2} + \frac{1}{\tan{\left(a \right)} + 1} 
                     2            2     
      1           sin (a)      sin (2*a)
------------- + ------------ + ---------
         2         2/    pi\    sin(4*a)
    2*sin (a)   sin |a + --|            
1 + ---------       \    2 /            
     sin(2*a)
sin⁡2(a)sin⁡2(a+π2)+sin⁡2(2a)sin⁡(4a)+12sin⁡2(a)sin⁡(2a)+1\frac{\sin^{2}{\left(a \right)}}{\sin^{2}{\left(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          1     
---------- + ------- + ----------
      1         2      2*cot(2*a)
1 + ------   cot (a)             
    cot(a)
12cot⁡(2a)+1cot⁡2(a)+11+1cot⁡(a)\frac{1}{2 \cot{\left(2 a \right)}} + \frac{1}{\cot^{2}{\left(a \right)}} + \frac{1}{1 + \frac{1}{\cot{\left(a \right)}}} 
                               2/a\   
                          4*tan |-|   
    1        tan(2*a)           \2/   
---------- + -------- + --------------
1 + tan(a)      2                    2
                        /       2/a\\ 
                        |1 - tan |-|| 
                        \        \2//
tan⁡(2a)2+1tan⁡(a)+1+4tan⁡2(a2)(1−tan⁡2(a2))2\frac{\tan{\left(2 a \right)}}{2} + \frac{1}{\tan{\left(a \right)} + 1} + \frac{4 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{2}} 
                   2         2     
      1         sin (a)   sin (2*a)
------------- + ------- + ---------
         2         2       sin(4*a)
    2*sin (a)   cos (a)            
1 + ---------                      
     sin(2*a)
sin⁡2(a)cos⁡2(a)+sin⁡2(2a)sin⁡(4a)+12sin⁡2(a)sin⁡(2a)+1\frac{\sin^{2}{\left(a \right)}}{\cos^{2}{\left(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} 
                                    /      pi\
                                 cos|2*a - --|
       1          1 - cos(2*a)      \      2 /
--------------- + ------------ + -------------
       /    pi\   1 + cos(2*a)     2*cos(2*a) 
    cos|a - --|                               
       \    2 /                               
1 + -----------                               
       cos(a)
1−cos⁡(2a)cos⁡(2a)+1+cos⁡(2a−π2)2cos⁡(2a)+11+cos⁡(a−π2)cos⁡(a)\frac{1 - \cos{\left(2 a \right)}}{\cos{\left(2 a \right)} + 1} + \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           2      tan(2*a)
---------- + tan (a) + --------
tan(a) + 1                2
tan⁡2(a)+tan⁡(2a)2+1tan⁡(a)+1\tan^{2}{\left(a \right)} + \frac{\tan{\left(2 a \right)}}{2} + \frac{1}{\tan{\left(a \right)} + 1} 
                   2             4   
      1         sin (2*a)   4*sin (a)
------------- + --------- + ---------
         2       sin(4*a)      2     
    2*sin (a)               sin (2*a)
1 + ---------                        
     sin(2*a)
4sin⁡4(a)sin⁡2(2a)+sin⁡2(2a)sin⁡(4a)+12sin⁡2(a)sin⁡(2a)+1\frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\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} 
    1        tan(2*a)   1 - cos(2*a)
---------- + -------- + ------------
1 + tan(a)      2       1 + cos(2*a)
1−cos⁡(2a)cos⁡(2a)+1+tan⁡(2a)2+1tan⁡(a)+1\frac{1 - \cos{\left(2 a \right)}}{\cos{\left(2 a \right)} + 1} + \frac{\tan{\left(2 a \right)}}{2} + \frac{1}{\tan{\left(a \right)} + 1} 
                            1                      
                  1 - -------------                
                         /pi      \      /pi      \
                      csc|-- - 2*a|   csc|-- - 2*a|
       1                 \2       /      \2       /
--------------- + ----------------- + -------------
       /pi    \             1           2*csc(2*a) 
    csc|-- - a|   1 + -------------                
       \2     /          /pi      \                
1 + -----------       csc|-- - 2*a|                
       csc(a)            \2       /
1−1csc⁡(−2a+π2)1+1csc⁡(−2a+π2)+csc⁡(−2a+π2)2csc⁡(2a)+11+csc⁡(−a+π2)csc⁡(a)\frac{1 - \frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}}}{1 + \frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}}} + \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)}}} 
                                   2   
                            1 - tan (a)
                        1 - -----------
                                   2   
    1        tan(2*a)       1 + tan (a)
---------- + -------- + ---------------
1 + tan(a)      2                  2   
                            1 - tan (a)
                        1 + -----------
                                   2   
                            1 + tan (a)
−1−tan⁡2(a)tan⁡2(a)+1+11−tan⁡2(a)tan⁡2(a)+1+1+tan⁡(2a)2+1tan⁡(a)+1\frac{- \frac{1 - \tan^{2}{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} + 1}{\frac{1 - \tan^{2}{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} + 1} + \frac{\tan{\left(2 a \right)}}{2} + \frac{1}{\tan{\left(a \right)} + 1} 
                2                
    1        sec (a)    sec(2*a) 
---------- + ------- + ----------
    sec(a)      2      2*csc(2*a)
1 + ------   csc (a)             
    csc(a)
sec⁡(2a)2csc⁡(2a)+sec⁡2(a)csc⁡2(a)+11+sec⁡(a)csc⁡(a)\frac{\sec{\left(2 a \right)}}{2 \csc{\left(2 a \right)}} + \frac{\sec^{2}{\left(a \right)}}{\csc^{2}{\left(a \right)}} + \frac{1}{1 + \frac{\sec{\left(a \right)}}{\csc{\left(a \right)}}} 
                2                
    1        sin (a)    sin(2*a) 
---------- + ------- + ----------
    sin(a)      2      2*cos(2*a)
1 + ------   cos (a)             
    cos(a)
sin⁡2(a)cos⁡2(a)+sin⁡(2a)2cos⁡(2a)+1sin⁡(a)cos⁡(a)+1\frac{\sin^{2}{\left(a \right)}}{\cos^{2}{\left(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} 
                     2/pi    \      /pi      \
                  csc |-- - a|   csc|-- - 2*a|
       1              \2     /      \2       /
--------------- + ------------ + -------------
       /pi    \        2           2*csc(2*a) 
    csc|-- - a|     csc (a)                   
       \2     /                               
1 + -----------                               
       csc(a)
csc⁡(−2a+π2)2csc⁡(2a)+csc⁡2(−a+π2)csc⁡2(a)+11+csc⁡(−a+π2)csc⁡(a)\frac{\csc{\left(- 2 a + \frac{\pi}{2} \right)}}{2 \csc{\left(2 a \right)}} + \frac{\csc^{2}{\left(- 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 + cot (a)
                          1 - ------------
                                     2    
    1            1            1 + cot (a) 
---------- + ---------- + ----------------
      1      2*cot(2*a)               2   
1 + ------                    -1 + cot (a)
    cot(a)                1 + ------------
                                     2    
                              1 + cot (a)
−cot⁡2(a)−1cot⁡2(a)+1+1cot⁡2(a)−1cot⁡2(a)+1+1+12cot⁡(2a)+11+1cot⁡(a)\frac{- \frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} + 1}{\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} + 1} + \frac{1}{2 \cot{\left(2 a \right)}} + \frac{1}{1 + \frac{1}{\cot{\left(a \right)}}} 
                       /pi      \            
                1 - sin|-- + 2*a|      2     
      1                \2       /   sin (2*a)
------------- + ----------------- + ---------
         2             /pi      \    sin(4*a)
    2*sin (a)   1 + sin|-- + 2*a|            
1 + ---------          \2       /            
     sin(2*a)
1−sin⁡(2a+π2)sin⁡(2a+π2)+1+sin⁡2(2a)sin⁡(4a)+12sin⁡2(a)sin⁡(2a)+1\frac{1 - \sin{\left(2 a + \frac{\pi}{2} \right)}}{\sin{\left(2 a + \frac{\pi}{2} \right)} + 1} + \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 - --------                  
       1              sec(2*a)       sec(2*a)   
--------------- + ------------ + ---------------
       sec(a)            1            /      pi\
1 + -----------   1 + --------   2*sec|2*a - --|
       /    pi\       sec(2*a)        \      2 /
    sec|a - --|                                 
       \    2 /
1−1sec⁡(2a)1+1sec⁡(2a)+sec⁡(2a)2sec⁡(2a−π2)+1sec⁡(a)sec⁡(a−π2)+1\frac{1 - \frac{1}{\sec{\left(2 a \right)}}}{1 + \frac{1}{\sec{\left(2 a \right)}}} + \frac{\sec{\left(2 a \right)}}{2 \sec{\left(2 a - \frac{\pi}{2} \right)}} + \frac{1}{\frac{\sec{\left(a \right)}}{\sec{\left(a - \frac{\pi}{2} \right)}} + 1} 
1/(1 + sec(a)/sec(a - pi/2)) + (1 - 1/sec(2*a))/(1 + 1/sec(2*a)) + sec(2*a)/(2*sec(2*a - pi/2))
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