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

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

Expresión a simplificar:

Solución

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