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Simplificación de expresiones/ Descomposición en fracciones simples/ cos(2*t)/(cos(t)+sin(t))

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

Expresión a simplificar:

Solución

Ha introducido [src] 
    cos(2*t)   
---------------
cos(t) + sin(t)
$$\frac{\cos{\left(2 t \right)}}{\sin{\left(t \right)} + \cos{\left(t \right)}}$$ 
cos(2*t)/(cos(t) + sin(t))
Simplificación general [src] 
  ___         
\/ 2 *cos(2*t)
--------------
     /    pi\ 
2*sin|t + --| 
     \    4 /
$$\frac{\sqrt{2} \cos{\left(2 t \right)}}{2 \sin{\left(t + \frac{\pi}{4} \right)}}$$ 
sqrt(2)*cos(2*t)/(2*sin(t + pi/4))
Respuesta numérica [src] 
cos(2*t)/(cos(t) + sin(t))
cos(2*t)/(cos(t) + sin(t))
Potencias [src] 
          -2*I*t    2*I*t        
         e         e             
         ------- + ------        
            2        2           
---------------------------------
 I*t    -I*t     /   -I*t    I*t\
e      e       I*\- e     + e   /
---- + ----- - ------------------
 2       2             2
$$\frac{\frac{e^{2 i t}}{2} + \frac{e^{- 2 i t}}{2}}{- \frac{i \left(e^{i t} - e^{- i t}\right)}{2} + \frac{e^{i t}}{2} + \frac{e^{- i t}}{2}}$$ 
(exp(-2*i*t)/2 + exp(2*i*t)/2)/(exp(i*t)/2 + exp(-i*t)/2 - i*(-exp(-i*t) + exp(i*t))/2)
Abrimos la expresión [src] 
                            2      
         1             2*cos (t)   
- --------------- + ---------------
  cos(t) + sin(t)   cos(t) + sin(t)
$$\frac{2 \cos^{2}{\left(t \right)}}{\sin{\left(t \right)} + \cos{\left(t \right)}} - \frac{1}{\sin{\left(t \right)} + \cos{\left(t \right)}}$$ 
-1/(cos(t) + sin(t)) + 2*cos(t)^2/(cos(t) + sin(t))
Parte trigonométrica [src] 
  ___             /    pi\
\/ 2 *cos(2*t)*csc|t + --|
                  \    4 /
--------------------------
            2
$$\frac{\sqrt{2} \cos{\left(2 t \right)} \csc{\left(t + \frac{\pi}{4} \right)}}{2}$$ 
  ___    /    pi\
\/ 2 *sec|t - --|
         \    4 /
-----------------
    2*sec(2*t)
$$\frac{\sqrt{2} \sec{\left(t - \frac{\pi}{4} \right)}}{2 \sec{\left(2 t \right)}}$$ 
                 1                  
------------------------------------
/  1           1     \    /pi      \
|------ + -----------|*csc|-- - 2*t|
|csc(t)      /pi    \|    \2       /
|         csc|-- - t||              
\            \2     //
$$\frac{1}{\left(\frac{1}{\csc{\left(- t + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(t \right)}}\right) \csc{\left(- 2 t + \frac{\pi}{2} \right)}}$$ 
                       2                  
               -1 + cot (t)               
------------------------------------------
              /        2/t\          /t\ \
              |-1 + cot |-|     2*cot|-| |
/       2   \ |         \2/          \2/ |
\1 + cot (t)/*|------------ + -----------|
              |       2/t\           2/t\|
              |1 + cot |-|    1 + cot |-||
              \        \2/            \2//
$$\frac{\cot^{2}{\left(t \right)} - 1}{\left(\frac{\cot^{2}{\left(\frac{t}{2} \right)} - 1}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{t}{2} \right)}}{\cot^{2}{\left(\frac{t}{2} \right)} + 1}\right) \left(\cot^{2}{\left(t \right)} + 1\right)}$$ 
      cos(2*t)      
--------------------
            /    pi\
cos(t) + cos|t - --|
            \    2 /
$$\frac{\cos{\left(2 t \right)}}{\cos{\left(t \right)} + \cos{\left(t - \frac{\pi}{2} \right)}}$$ 
      /pi      \    
   sin|-- + 2*t|    
      \2       /    
--------------------
            /    pi\
sin(t) + sin|t + --|
            \    2 /
$$\frac{\sin{\left(2 t + \frac{\pi}{2} \right)}}{\sin{\left(t \right)} + \sin{\left(t + \frac{\pi}{2} \right)}}$$ 
  ___             /    pi\
\/ 2 *cos(2*t)*sec|t - --|
                  \    4 /
--------------------------
            2
$$\frac{\sqrt{2} \cos{\left(2 t \right)} \sec{\left(t - \frac{\pi}{4} \right)}}{2}$$ 
  ___    2/t   pi\ /        2/t   pi\\
\/ 2 *sin |- + --|*|-1 + cot |- + --||
          \2   8 / \         \2   8 //
$$\sqrt{2} \left(\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)} - 1\right) \sin^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}$$ 
  ___ /       2/t   pi\\ /       2   \
\/ 2 *|1 + tan |- + --||*\1 - tan (t)/
      \        \2   8 //              
--------------------------------------
       /       2   \    /t   pi\      
     4*\1 + tan (t)/*tan|- + --|      
                        \2   8 /
$$\frac{\sqrt{2} \left(1 - \tan^{2}{\left(t \right)}\right) \left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)} + 1\right)}{4 \left(\tan^{2}{\left(t \right)} + 1\right) \tan{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}$$ 
  ___    /pi      \
\/ 2 *sin|-- + 2*t|
         \2       /
-------------------
        /    pi\   
   2*sin|t + --|   
        \    4 /
$$\frac{\sqrt{2} \sin{\left(2 t + \frac{\pi}{2} \right)}}{2 \sin{\left(t + \frac{\pi}{4} \right)}}$$ 
  ___ /       2/t   pi\\ /        2   \
\/ 2 *|1 + cot |- + --||*\-1 + cot (t)/
      \        \2   8 //               
---------------------------------------
        /       2   \    /t   pi\      
      4*\1 + cot (t)/*cot|- + --|      
                         \2   8 /
$$\frac{\sqrt{2} \left(\cot^{2}{\left(t \right)} - 1\right) \left(\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)} + 1\right)}{4 \left(\cot^{2}{\left(t \right)} + 1\right) \cot{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}$$ 
  ___    2    /       2/t   pi\\ /       2   \
\/ 2 *cos (t)*|1 + tan |- + --||*\1 - tan (t)/
              \        \2   8 //              
----------------------------------------------
                     /t   pi\                 
                4*tan|- + --|                 
                     \2   8 /
$$\frac{\sqrt{2} \left(1 - \tan^{2}{\left(t \right)}\right) \left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)} + 1\right) \cos^{2}{\left(t \right)}}{4 \tan{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}$$ 
     ___   
   \/ 2    
-----------
   /    pi\
sec|t + --|
   \    4 /
$$\frac{\sqrt{2}}{\sec{\left(t + \frac{\pi}{4} \right)}}$$ 
            1             
--------------------------
/  1        1   \         
|------ + ------|*sec(2*t)
\csc(t)   sec(t)/
$$\frac{1}{\left(\frac{1}{\sec{\left(t \right)}} + \frac{1}{\csc{\left(t \right)}}\right) \sec{\left(2 t \right)}}$$ 
                      2                  
               1 - tan (t)               
-----------------------------------------
              /       2/t\          /t\ \
              |1 - tan |-|     2*tan|-| |
/       2   \ |        \2/          \2/ |
\1 + tan (t)/*|----------- + -----------|
              |       2/t\          2/t\|
              |1 + tan |-|   1 + tan |-||
              \        \2/           \2//
$$\frac{1 - \tan^{2}{\left(t \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{t}{2} \right)}}{\tan^{2}{\left(\frac{t}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{t}{2} \right)}}{\tan^{2}{\left(\frac{t}{2} \right)} + 1}\right) \left(\tan^{2}{\left(t \right)} + 1\right)}$$ 
  ___ /       2/t   pi\\
\/ 2 *|1 - tan |- + --||
      \        \2   8 //
------------------------
           2/t   pi\    
    1 + tan |- + --|    
            \2   8 /
$$\frac{\sqrt{2} \left(1 - \tan^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}\right)}{\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)} + 1}$$ 
  ___ /        2/t   pi\\
\/ 2 *|-1 + cot |- + --||
      \         \2   8 //
-------------------------
            2/t   pi\    
     1 + cot |- + --|    
             \2   8 /
$$\frac{\sqrt{2} \left(\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)} - 1\right)}{\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)} + 1}$$ 
  ___    /    3*pi\
\/ 2 *sin|t + ----|
         \     4  /
$$\sqrt{2} \sin{\left(t + \frac{3 \pi}{4} \right)}$$ 
     ___    
   \/ 2     
------------
   /     pi\
csc|-t + --|
   \     4 /
$$\frac{\sqrt{2}}{\csc{\left(- t + \frac{\pi}{4} \right)}}$$ 
  ___         
\/ 2 *cos(2*t)
--------------
     /    pi\ 
2*cos|t - --| 
     \    4 /
$$\frac{\sqrt{2} \cos{\left(2 t \right)}}{2 \cos{\left(t - \frac{\pi}{4} \right)}}$$ 
  ___    2    /        2   \    /t   pi\
\/ 2 *sin (t)*\-1 + cot (t)/*tan|- + --|
                                \2   8 /
----------------------------------------
                  2/t   pi\             
             4*sin |- + --|             
                   \2   8 /
$$\frac{\sqrt{2} \left(\cot^{2}{\left(t \right)} - 1\right) \sin^{2}{\left(t \right)} \tan{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}{4 \sin^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}$$ 
  ___    /    pi\
\/ 2 *csc|t + --|
         \    4 /
-----------------
      /pi      \ 
 2*csc|-- - 2*t| 
      \2       /
$$\frac{\sqrt{2} \csc{\left(t + \frac{\pi}{4} \right)}}{2 \csc{\left(- 2 t + \frac{\pi}{2} \right)}}$$ 
               1               
-------------------------------
/  1           1     \         
|------ + -----------|*sec(2*t)
|sec(t)      /    pi\|         
|         sec|t - --||         
\            \    2 //
$$\frac{1}{\left(\frac{1}{\sec{\left(t - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(t \right)}}\right) \sec{\left(2 t \right)}}$$ 
  ___    /    pi\
\/ 2 *cos|t + --|
         \    4 /
$$\sqrt{2} \cos{\left(t + \frac{\pi}{4} \right)}$$ 
  ___         
\/ 2 *cos(2*t)
--------------
     /    pi\ 
2*sin|t + --| 
     \    4 /
$$\frac{\sqrt{2} \cos{\left(2 t \right)}}{2 \sin{\left(t + \frac{\pi}{4} \right)}}$$ 
sqrt(2)*cos(2*t)/(2*sin(t + pi/4))
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