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Simplificación de expresiones/ Descomposición en fracciones simples/ tan(x)+coscos(x)/(sin*sin(x)+1)

¿Cómo vas a descomponer esta tan(x)+coscos(x)/(sin*sin(x)+1) expresión en fracciones?

Expresión a simplificar:

Solución

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