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Simplificación de expresiones/ Descomposición en fracciones simples/ sqrt(2)*(2+cot(x)^2+tan(x)^2)/(2*(1+(tan(x)-cot(x))^2/2))

¿Cómo vas a descomponer esta sqrt(2)*(2+cot(x)^2+tan(x)^2)/(2*(1+(tan(x)-cot(x))^2/2)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
  ___ /       2         2   \
\/ 2 *\2 + cot (x) + tan (x)/
-----------------------------
    /                     2\ 
    |    (tan(x) - cot(x)) | 
  2*|1 + ------------------| 
    \            2         /
$$\frac{\sqrt{2} \left(\left(\cot^{2}{\left(x \right)} + 2\right) + \tan^{2}{\left(x \right)}\right)}{2 \left(\frac{\left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2}}{2} + 1\right)}$$ 
(sqrt(2)*(2 + cot(x)^2 + tan(x)^2))/((2*(1 + (tan(x) - cot(x))^2/2)))
Simplificación general [src] 
                   2
  ___ /       2   \ 
\/ 2 *\1 + tan (x)/ 
--------------------
           4        
    1 + tan (x)
$$\frac{\sqrt{2} \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{4}{\left(x \right)} + 1}$$ 
sqrt(2)*(1 + tan(x)^2)^2/(1 + tan(x)^4)
Respuesta numérica [src] 
(2.82842712474619 + 1.4142135623731*cot(x)^2 + 1.4142135623731*tan(x)^2)/(2.0 + 1.0*(-cot(x) + tan(x))^2)
(2.82842712474619 + 1.4142135623731*cot(x)^2 + 1.4142135623731*tan(x)^2)/(2.0 + 1.0*(-cot(x) + tan(x))^2)
Potencias [src] 
  ___ /       2         2   \
\/ 2 *\2 + cot (x) + tan (x)/
-----------------------------
                         2   
   2 + (-cot(x) + tan(x))
$$\frac{\sqrt{2} \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 2\right)}{\left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2} + 2}$$ 
      /                              2\
      |              /   I*x    -I*x\ |
  ___ |       2      \- e    + e    / |
\/ 2 *|2 + cot (x) - -----------------|
      |                             2 |
      |               / I*x    -I*x\  |
      \               \e    + e    /  /
---------------------------------------
                                    2  
      /            /   I*x    -I*x\\   
      |          I*\- e    + e    /|   
  2 + |-cot(x) + ------------------|   
      |              I*x    -I*x   |   
      \             e    + e       /
$$\frac{\sqrt{2} \left(- \frac{\left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} + \cot^{2}{\left(x \right)} + 2\right)}{\left(\frac{i \left(- e^{i x} + e^{- i x}\right)}{e^{i x} + e^{- i x}} - \cot{\left(x \right)}\right)^{2} + 2}$$ 
sqrt(2)*(2 + cot(x)^2 - (-exp(i*x) + exp(-i*x))^2/(exp(i*x) + exp(-i*x))^2)/(2 + (-cot(x) + i*(-exp(i*x) + exp(-i*x))/(exp(i*x) + exp(-i*x)))^2)
Combinatoria [src] 
       ___ /       2         2   \     
     \/ 2 *\2 + cot (x) + tan (x)/     
---------------------------------------
       2         2                     
2 + cot (x) + tan (x) - 2*cot(x)*tan(x)
$$\frac{\sqrt{2} \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 2\right)}{\tan^{2}{\left(x \right)} - 2 \tan{\left(x \right)} \cot{\left(x \right)} + \cot^{2}{\left(x \right)} + 2}$$ 
sqrt(2)*(2 + cot(x)^2 + tan(x)^2)/(2 + cot(x)^2 + tan(x)^2 - 2*cot(x)*tan(x))
Unión de expresiones racionales [src] 
  ___ /       2         2   \
\/ 2 *\2 + cot (x) + tan (x)/
-----------------------------
                         2   
   2 + (-cot(x) + tan(x))
$$\frac{\sqrt{2} \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 2\right)}{\left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2} + 2}$$ 
sqrt(2)*(2 + cot(x)^2 + tan(x)^2)/(2 + (-cot(x) + tan(x))^2)
Denominador común [src] 
                     ___                       
  ___            2*\/ 2 *cot(x)*tan(x)         
\/ 2  + ---------------------------------------
               2         2                     
        2 + cot (x) + tan (x) - 2*cot(x)*tan(x)
$$\sqrt{2} + \frac{2 \sqrt{2} \tan{\left(x \right)} \cot{\left(x \right)}}{\tan^{2}{\left(x \right)} - 2 \tan{\left(x \right)} \cot{\left(x \right)} + \cot^{2}{\left(x \right)} + 2}$$ 
sqrt(2) + 2*sqrt(2)*cot(x)*tan(x)/(2 + cot(x)^2 + tan(x)^2 - 2*cot(x)*tan(x))
Abrimos la expresión [src] 
                    ___                                  ___    2                                  ___    2                
                2*\/ 2                                 \/ 2 *cot (x)                             \/ 2 *tan (x)             
--------------------------------------- + --------------------------------------- + ---------------------------------------
       2         2                               2         2                               2         2                     
2 + cot (x) + tan (x) - 2*cot(x)*tan(x)   2 + cot (x) + tan (x) - 2*cot(x)*tan(x)   2 + cot (x) + tan (x) - 2*cot(x)*tan(x)
$$\frac{\sqrt{2} \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} - 2 \tan{\left(x \right)} \cot{\left(x \right)} + \cot^{2}{\left(x \right)} + 2} + \frac{\sqrt{2} \cot^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} - 2 \tan{\left(x \right)} \cot{\left(x \right)} + \cot^{2}{\left(x \right)} + 2} + \frac{2 \sqrt{2}}{\tan^{2}{\left(x \right)} - 2 \tan{\left(x \right)} \cot{\left(x \right)} + \cot^{2}{\left(x \right)} + 2}$$ 
2*sqrt(2)/(2 + cot(x)^2 + tan(x)^2 - 2*cot(x)*tan(x)) + sqrt(2)*cot(x)^2/(2 + cot(x)^2 + tan(x)^2 - 2*cot(x)*tan(x)) + sqrt(2)*tan(x)^2/(2 + cot(x)^2 + tan(x)^2 - 2*cot(x)*tan(x))
Compilar la expresión [src] 
  ___ /       2         2   \
\/ 2 *\2 + cot (x) + tan (x)/
-----------------------------
                         2   
   2 + (-cot(x) + tan(x))
$$\frac{\sqrt{2} \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 2\right)}{\left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2} + 2}$$ 
sqrt(2)*(2 + cot(x)^2 + tan(x)^2)/(2 + (-cot(x) + tan(x))^2)
Denominador racional [src] 
       ___ /       2         2   \     
     \/ 2 *\2 + cot (x) + tan (x)/     
---------------------------------------
       2         2                     
2 + cot (x) + tan (x) - 2*cot(x)*tan(x)
$$\frac{\sqrt{2} \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 2\right)}{\tan^{2}{\left(x \right)} - 2 \tan{\left(x \right)} \cot{\left(x \right)} + \cot^{2}{\left(x \right)} + 2}$$ 
sqrt(2)*(2 + cot(x)^2 + tan(x)^2)/(2 + cot(x)^2 + tan(x)^2 - 2*cot(x)*tan(x))
Parte trigonométrica [src] 
  ___ /       1         2   \
\/ 2 *|2 + ------- + cot (x)|
      |       2             |
      \    cot (x)          /
-----------------------------
                         2   
        /  1            \    
    2 + |------ - cot(x)|    
        \cot(x)         /
$$\frac{\sqrt{2} \left(\cot^{2}{\left(x \right)} + 2 + \frac{1}{\cot^{2}{\left(x \right)}}\right)}{\left(- \cot{\left(x \right)} + \frac{1}{\cot{\left(x \right)}}\right)^{2} + 2}$$ 
                        2
      /       2/pi    \\ 
      |    csc |-- - x|| 
  ___ |        \2     /| 
\/ 2 *|1 + ------------| 
      |         2      | 
      \      csc (x)   / 
-------------------------
            4/pi    \    
         csc |-- - x|    
             \2     /    
     1 + ------------    
              4          
           csc (x)
$$\frac{\sqrt{2} \left(1 + \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right)^{2}}{1 + \frac{\csc^{4}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{4}{\left(x \right)}}}$$ 
  ___ /       1         2   \
\/ 2 *|2 + ------- + tan (x)|
      |       2             |
      \    tan (x)          /
-----------------------------
                          2  
       /    1            \   
   2 + |- ------ + tan(x)|   
       \  tan(x)         /
$$\frac{\sqrt{2} \left(\tan^{2}{\left(x \right)} + 2 + \frac{1}{\tan^{2}{\left(x \right)}}\right)}{\left(\tan{\left(x \right)} - \frac{1}{\tan{\left(x \right)}}\right)^{2} + 2}$$ 
                   2
  ___ /       2   \ 
\/ 2 *\1 + tan (x)/ 
--------------------
           4        
    1 + tan (x)
$$\frac{\sqrt{2} \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{4}{\left(x \right)} + 1}$$ 
               ___             
             \/ 2              
-------------------------------
             4                 
/       2/x\\            4/x\  
|1 - tan |-||      16*tan |-|  
\        \2//             \2/  
-------------- + --------------
             4                4
/       2/x\\    /       2/x\\ 
|1 + tan |-||    |1 + tan |-|| 
\        \2//    \        \2//
$$\frac{\sqrt{2}}{\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{4}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} + \frac{16 \tan^{4}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}}}$$ 
      /       2         2   \
  ___ |    sin (x)   cos (x)|
\/ 2 *|2 + ------- + -------|
      |       2         2   |
      \    cos (x)   sin (x)/
-----------------------------
                         2   
        /sin(x)   cos(x)\    
    2 + |------ - ------|    
        \cos(x)   sin(x)/
$$\frac{\sqrt{2} \left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right)}{\left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\right)^{2} + 2}$$ 
        ___      
      \/ 2       
-----------------
   4         4   
cos (x) + sin (x)
$$\frac{\sqrt{2}}{\sin^{4}{\left(x \right)} + \cos^{4}{\left(x \right)}}$$ 
          ___         
        \/ 2          
----------------------
   4         4/    pi\
sin (x) + sin |x + --|
              \    2 /
$$\frac{\sqrt{2}}{\sin^{4}{\left(x \right)} + \sin^{4}{\left(x + \frac{\pi}{2} \right)}}$$ 
          ___         
        \/ 2          
----------------------
   1           1      
------- + ------------
   4         4/pi    \
csc (x)   csc |-- - x|
              \2     /
$$\frac{\sqrt{2}}{\frac{1}{\csc^{4}{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{4}{\left(x \right)}}}$$ 
                   2
  ___ /       1   \ 
\/ 2 *|1 + -------| 
      |       2   | 
      \    cot (x)/ 
--------------------
           1        
    1 + -------     
           4        
        cot (x)
$$\frac{\sqrt{2} \left(1 + \frac{1}{\cot^{2}{\left(x \right)}}\right)^{2}}{1 + \frac{1}{\cot^{4}{\left(x \right)}}}$$ 
      /       2/    pi\               \
      |    cos |x - --|        2      |
  ___ |        \    2 /     cos (x)   |
\/ 2 *|2 + ------------ + ------------|
      |         2            2/    pi\|
      |      cos (x)      cos |x - --||
      \                       \    2 //
---------------------------------------
                                   2   
        /   /    pi\              \    
        |cos|x - --|              |    
        |   \    2 /      cos(x)  |    
    2 + |----------- - -----------|    
        |   cos(x)        /    pi\|    
        |              cos|x - --||    
        \                 \    2 //
$$\frac{\sqrt{2} \left(\frac{\cos^{2}{\left(x \right)}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} + 2 + \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right)}{\left(- \frac{\cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}\right)^{2} + 2}$$ 
      /       2         2   \
  ___ |    sec (x)   csc (x)|
\/ 2 *|2 + ------- + -------|
      |       2         2   |
      \    csc (x)   sec (x)/
-----------------------------
                         2   
        /sec(x)   csc(x)\    
    2 + |------ - ------|    
        \csc(x)   sec(x)/
$$\frac{\sqrt{2} \left(\frac{\csc^{2}{\left(x \right)}}{\sec^{2}{\left(x \right)}} + 2 + \frac{\sec^{2}{\left(x \right)}}{\csc^{2}{\left(x \right)}}\right)}{\left(- \frac{\csc{\left(x \right)}}{\sec{\left(x \right)}} + \frac{\sec{\left(x \right)}}{\csc{\left(x \right)}}\right)^{2} + 2}$$ 
                     ___                   
                   \/ 2                    
-------------------------------------------
             4                             
/       2/x\\     8/x\         8/x\    4/x\
|1 - tan |-|| *cos |-| + 16*cos |-|*tan |-|
\        \2//      \2/          \2/     \2/
$$\frac{\sqrt{2}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{4} \cos^{8}{\left(\frac{x}{2} \right)} + 16 \cos^{8}{\left(\frac{x}{2} \right)} \tan^{4}{\left(\frac{x}{2} \right)}}$$ 
                        2
      /       2/    pi\\ 
      |    cos |x - --|| 
  ___ |        \    2 /| 
\/ 2 *|1 + ------------| 
      |         2      | 
      \      cos (x)   / 
-------------------------
            4/    pi\    
         cos |x - --|    
             \    2 /    
     1 + ------------    
              4          
           cos (x)
$$\frac{\sqrt{2} \left(1 + \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right)^{2}}{1 + \frac{\cos^{4}{\left(x - \frac{\pi}{2} \right)}}{\cos^{4}{\left(x \right)}}}$$ 
                        2
      /         2      \ 
  ___ |      sec (x)   | 
\/ 2 *|1 + ------------| 
      |       2/    pi\| 
      |    sec |x - --|| 
      \        \    2 // 
-------------------------
              4          
           sec (x)       
     1 + ------------    
            4/    pi\    
         sec |x - --|    
             \    2 /
$$\frac{\sqrt{2} \left(\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right)^{2}}{\frac{\sec^{4}{\left(x \right)}}{\sec^{4}{\left(x - \frac{\pi}{2} \right)}} + 1}$$ 
                     2
      /         4   \ 
  ___ |    4*sin (x)| 
\/ 2 *|1 + ---------| 
      |       2     | 
      \    sin (2*x)/ 
----------------------
              8       
        16*sin (x)    
    1 + ----------    
           4          
        sin (2*x)
$$\frac{\sqrt{2} \left(\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1\right)^{2}}{\frac{16 \sin^{8}{\left(x \right)}}{\sin^{4}{\left(2 x \right)}} + 1}$$ 
          ___        
        \/ 2         
---------------------
/       4   \    4   
\1 + tan (x)/*cos (x)
$$\frac{\sqrt{2}}{\left(\tan^{4}{\left(x \right)} + 1\right) \cos^{4}{\left(x \right)}}$$ 
                     ___                    
                   \/ 2                     
--------------------------------------------
              4                             
/        2/x\\     8/x\         4/x\    8/x\
|-1 + cot |-|| *sin |-| + 16*cot |-|*sin |-|
\         \2//      \2/          \2/     \2/
$$\frac{\sqrt{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)} + 16 \sin^{8}{\left(\frac{x}{2} \right)} \cot^{4}{\left(\frac{x}{2} \right)}}$$ 
          ___         
        \/ 2          
----------------------
   4         4/    pi\
cos (x) + cos |x - --|
              \    2 /
$$\frac{\sqrt{2}}{\cos^{4}{\left(x \right)} + \cos^{4}{\left(x - \frac{\pi}{2} \right)}}$$ 
      /         4         2     \
  ___ |    4*sin (x)   sin (2*x)|
\/ 2 *|2 + --------- + ---------|
      |       2             4   |
      \    sin (2*x)   4*sin (x)/
---------------------------------
                              2  
       /     2               \   
       |2*sin (x)    sin(2*x)|   
   2 + |--------- - ---------|   
       | sin(2*x)        2   |   
       \            2*sin (x)/
$$\frac{\sqrt{2} \left(\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 2 + \frac{\sin^{2}{\left(2 x \right)}}{4 \sin^{4}{\left(x \right)}}\right)}{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} - \frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(x \right)}}\right)^{2} + 2}$$ 
      /       2/pi    \               \
      |    csc |-- - x|        2      |
  ___ |        \2     /     csc (x)   |
\/ 2 *|2 + ------------ + ------------|
      |         2            2/pi    \|
      |      csc (x)      csc |-- - x||
      \                       \2     //
---------------------------------------
                                   2   
        /   /pi    \              \    
        |csc|-- - x|              |    
        |   \2     /      csc(x)  |    
    2 + |----------- - -----------|    
        |   csc(x)        /pi    \|    
        |              csc|-- - x||    
        \                 \2     //
$$\frac{\sqrt{2} \left(\frac{\csc^{2}{\left(x \right)}}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + 2 + \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right)}{\left(- \frac{\csc{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}\right)^{2} + 2}$$ 
          ___         
        \/ 2          
----------------------
   1           1      
------- + ------------
   4         4/    pi\
sec (x)   sec |x - --|
              \    2 /
$$\frac{\sqrt{2}}{\frac{1}{\sec^{4}{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec^{4}{\left(x \right)}}}$$ 
      /       2/    pi\               \
      |    sec |x - --|        2      |
  ___ |        \    2 /     sec (x)   |
\/ 2 *|2 + ------------ + ------------|
      |         2            2/    pi\|
      |      sec (x)      sec |x - --||
      \                       \    2 //
---------------------------------------
                                   2   
        /                 /    pi\\    
        |              sec|x - --||    
        |   sec(x)        \    2 /|    
    2 + |----------- - -----------|    
        |   /    pi\      sec(x)  |    
        |sec|x - --|              |    
        \   \    2 /              /
$$\frac{\sqrt{2} \left(\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 2 + \frac{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(x \right)}}\right)}{\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} - \frac{\sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}}\right)^{2} + 2}$$ 
               ___              
             \/ 2               
--------------------------------
              4                 
/        2/x\\            4/x\  
|-1 + cot |-||      16*cot |-|  
\         \2//             \2/  
--------------- + --------------
              4                4
 /       2/x\\    /       2/x\\ 
 |1 + cot |-||    |1 + cot |-|| 
 \        \2//    \        \2//
$$\frac{\sqrt{2}}{\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} + \frac{16 \cot^{4}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}}}$$ 
sqrt(2)/((-1 + cot(x/2)^2)^4/(1 + cot(x/2)^2)^4 + 16*cot(x/2)^4/(1 + cot(x/2)^2)^4)
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