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Simplificación de expresiones/ Descomposición en fracciones simples/ cos(x)/(cos(x+pi/6))

¿Cómo vas a descomponer esta cos(x)/(cos(x+pi/6)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
   cos(x)  
-----------
   /    pi\
cos|x + --|
   \    6 /
cos⁡(x)cos⁡(x+π6)\frac{\cos{\left(x \right)}}{\cos{\left(x + \frac{\pi}{6} \right)}} 
cos(x)/cos(x + pi/6)
Respuesta numérica [src] 
cos(x)/cos(x + pi/6)
cos(x)/cos(x + pi/6)
Potencias [src] 
        I*x    -I*x       
       e      e           
       ---- + -----       
        2       2         
--------------------------
   /    pi\      /     pi\
 I*|x + --|    I*|-x - --|
   \    6 /      \     6 /
e             e           
----------- + ------------
     2             2
eix2+e−ix2ei(−x−π6)2+ei(x+π6)2\frac{\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}}{\frac{e^{i \left(- x - \frac{\pi}{6}\right)}}{2} + \frac{e^{i \left(x + \frac{\pi}{6}\right)}}{2}} 
(exp(i*x)/2 + exp(-i*x)/2)/(exp(i*(x + pi/6))/2 + exp(i*(-x - pi/6))/2)
Unión de expresiones racionales [src] 
    cos(x)   
-------------
   /pi + 6*x\
cos|--------|
   \   6    /
cos⁡(x)cos⁡(6x+π6)\frac{\cos{\left(x \right)}}{\cos{\left(\frac{6 x + \pi}{6} \right)}} 
cos(x)/cos((pi + 6*x)/6)
Abrimos la expresión [src] 
         cos(x)        
-----------------------
             ___       
  sin(x)   \/ 3 *cos(x)
- ------ + ------------
    2           2
cos⁡(x)−sin⁡(x)2+3cos⁡(x)2\frac{\cos{\left(x \right)}}{- \frac{\sin{\left(x \right)}}{2} + \frac{\sqrt{3} \cos{\left(x \right)}}{2}} 
cos(x)/(-sin(x)/2 + sqrt(3)*cos(x)/2)
Parte trigonométrica [src] 
          /    pi\
cos(x)*sec|x + --|
          \    6 /
cos⁡(x)sec⁡(x+π6)\cos{\left(x \right)} \sec{\left(x + \frac{\pi}{6} \right)} 
              2                                 
/        2/x\\  /         1      \ /       1   \
|-1 + cot |-|| *|1 + ------------|*|1 - -------|
\         \4//  |       2/x   pi\| |       2/x\|
                |    cot |- + --|| |    cot |-||
                \        \2   12// \        \2//
------------------------------------------------
                    2                           
       /       2/x\\  /         1      \        
       |1 + cot |-|| *|1 - ------------|        
       \        \4//  |       2/x   pi\|        
                      |    cot |- + --||        
                      \        \2   12//
(1−1cot⁡2(x2))(1+1cot⁡2(x2+π12))(cot⁡2(x4)−1)2(1−1cot⁡2(x2+π12))(cot⁡2(x4)+1)2\frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{12} \right)}}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{2}}{\left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{12} \right)}}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}} 
             /         4/x\\ /         4/x   pi\\
             |    4*sin |-|| |    4*sin |- + --||
   2/pi   x\ |          \2/| |          \2   12/|
sin |-- + -|*|1 - ---------|*|1 + --------------|
    \2    2/ |        2    | |        2/    pi\ |
             \     sin (x) / |     sin |x + --| |
                             \         \    6 / /
-------------------------------------------------
                         4/x   pi\               
                    4*sin |- + --|               
                          \2   12/               
                1 - --------------               
                        2/    pi\                
                     sin |x + --|                
                         \    6 /
(−4sin⁡4(x2)sin⁡2(x)+1)(4sin⁡4(x2+π12)sin⁡2(x+π6)+1)sin⁡2(x2+π2)−4sin⁡4(x2+π12)sin⁡2(x+π6)+1\frac{\left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{12} \right)}}{\sin^{2}{\left(x + \frac{\pi}{6} \right)}} + 1\right) \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{- \frac{4 \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{12} \right)}}{\sin^{2}{\left(x + \frac{\pi}{6} \right)}} + 1} 
              2                                         
/        2/x\\     4/x\ /       2/x   pi\\ /       2/x\\
|-1 + cot |-|| *sin |-|*|1 + tan |- + --||*|1 - tan |-||
\         \4//      \4/ \        \2   12// \        \2//
--------------------------------------------------------
                           2/x   pi\                    
                    1 - tan |- + --|                    
                            \2   12/
(1−tan⁡2(x2))(tan⁡2(x2+π12)+1)(cot⁡2(x4)−1)2sin⁡4(x4)1−tan⁡2(x2+π12)\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{12} \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{x}{4} \right)}}{1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{12} \right)}} 
   /    pi\
sec|x + --|
   \    6 /
-----------
   sec(x)
sec⁡(x+π6)sec⁡(x)\frac{\sec{\left(x + \frac{\pi}{6} \right)}}{\sec{\left(x \right)}} 
        /       2/x   5*pi\\ /       2/x   pi\\
        |    cos |- - ----|| |    cos |- - --||
   2/x\ |        \2    12 /| |        \2   2 /|
cos |-|*|1 + --------------|*|1 - ------------|
    \2/ |        2/x   pi\ | |         2/x\   |
        |     cos |- + --| | |      cos |-|   |
        \         \2   12/ / \          \2/   /
-----------------------------------------------
                      2/x   5*pi\              
                   cos |- - ----|              
                       \2    12 /              
               1 - --------------              
                       2/x   pi\               
                    cos |- + --|               
                        \2   12/
(1−cos⁡2(x2−π2)cos⁡2(x2))(cos⁡2(x2−5π12)cos⁡2(x2+π12)+1)cos⁡2(x2)−cos⁡2(x2−5π12)cos⁡2(x2+π12)+1\frac{\left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \left(\frac{\cos^{2}{\left(\frac{x}{2} - \frac{5 \pi}{12} \right)}}{\cos^{2}{\left(\frac{x}{2} + \frac{\pi}{12} \right)}} + 1\right) \cos^{2}{\left(\frac{x}{2} \right)}}{- \frac{\cos^{2}{\left(\frac{x}{2} - \frac{5 \pi}{12} \right)}}{\cos^{2}{\left(\frac{x}{2} + \frac{\pi}{12} \right)}} + 1} 
          /          2/x   pi\\       
          |     2*sin |- + --||       
     2/x\ |           \2   12/|       
2*cos |-|*|1 + ---------------|*cos(x)
      \2/ |           /    pi\|       
          |    1 + cos|x + --||       
          \           \    6 //       
--------------------------------------
  /          2/x   pi\\               
  |     2*sin |- + --||               
  |           \2   12/|               
  |1 - ---------------|*(1 + cos(x))  
  |           /    pi\|               
  |    1 + cos|x + --||               
  \           \    6 //
2(1+2sin⁡2(x2+π12)cos⁡(x+π6)+1)cos⁡2(x2)cos⁡(x)(1−2sin⁡2(x2+π12)cos⁡(x+π6)+1)(cos⁡(x)+1)\frac{2 \left(1 + \frac{2 \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{12} \right)}}{\cos{\left(x + \frac{\pi}{6} \right)} + 1}\right) \cos^{2}{\left(\frac{x}{2} \right)} \cos{\left(x \right)}}{\left(1 - \frac{2 \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{12} \right)}}{\cos{\left(x + \frac{\pi}{6} \right)} + 1}\right) \left(\cos{\left(x \right)} + 1\right)} 
   2/x\ /       2/x   pi\\ /        2/x\\
sin |-|*|1 + tan |- + --||*|-1 + cot |-||
    \2/ \        \2   3 // \         \2//
-----------------------------------------
                   /x   pi\              
              2*tan|- + --|              
                   \2   3 /
(tan⁡2(x2+π3)+1)(cot⁡2(x2)−1)sin⁡2(x2)2tan⁡(x2+π3)\frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{3} \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \sin^{2}{\left(\frac{x}{2} \right)}}{2 \tan{\left(\frac{x}{2} + \frac{\pi}{3} \right)}} 
   /     pi\
csc|-x + --|
   \     3 /
------------
   /pi    \ 
csc|-- - x| 
   \2     /
csc⁡(−x+π3)csc⁡(−x+π2)\frac{\csc{\left(- x + \frac{\pi}{3} \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}} 
          /     pi\
cos(x)*csc|-x + --|
          \     3 /
cos⁡(x)csc⁡(−x+π3)\cos{\left(x \right)} \csc{\left(- x + \frac{\pi}{3} \right)} 
     2/x\ /                  2               \       
2*cos |-|*|1 + ------------------------------|*cos(x)
      \2/ |    /       /    pi\\    2/x   pi\|       
          |    |1 + cos|x + --||*csc |- + --||       
          \    \       \    6 //     \2   12//       
-----------------------------------------------------
  /                  2               \               
  |1 - ------------------------------|*(1 + cos(x))  
  |    /       /    pi\\    2/x   pi\|               
  |    |1 + cos|x + --||*csc |- + --||               
  \    \       \    6 //     \2   12//
2(1+2(cos⁡(x+π6)+1)csc⁡2(x2+π12))cos⁡2(x2)cos⁡(x)(1−2(cos⁡(x+π6)+1)csc⁡2(x2+π12))(cos⁡(x)+1)\frac{2 \left(1 + \frac{2}{\left(\cos{\left(x + \frac{\pi}{6} \right)} + 1\right) \csc^{2}{\left(\frac{x}{2} + \frac{\pi}{12} \right)}}\right) \cos^{2}{\left(\frac{x}{2} \right)} \cos{\left(x \right)}}{\left(1 - \frac{2}{\left(\cos{\left(x + \frac{\pi}{6} \right)} + 1\right) \csc^{2}{\left(\frac{x}{2} + \frac{\pi}{12} \right)}}\right) \left(\cos{\left(x \right)} + 1\right)} 
   /    pi\
sec|x + --|
   \    6 /
-----------
   /pi    \
csc|-- - x|
   \2     /
sec⁡(x+π6)csc⁡(−x+π2)\frac{\sec{\left(x + \frac{\pi}{6} \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}} 
/       2/x   pi\\ /        2/x\\
|1 + cot |- + --||*|-1 + cot |-||
\        \2   12// \         \2//
---------------------------------
/       2/x\\ /        2/x   pi\\
|1 + cot |-||*|-1 + cot |- + --||
\        \2// \         \2   12//
(cot⁡2(x2)−1)(cot⁡2(x2+π12)+1)(cot⁡2(x2)+1)(cot⁡2(x2+π12)−1)\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{12} \right)} + 1\right)}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{12} \right)} - 1\right)} 
   /    pi\
sin|x + --|
   \    2 /
-----------
   /    pi\
cos|x + --|
   \    6 /
sin⁡(x+π2)cos⁡(x+π6)\frac{\sin{\left(x + \frac{\pi}{2} \right)}}{\cos{\left(x + \frac{\pi}{6} \right)}} 
/        2/x   pi\ \ /         2/x\   \
|     sec |- + --| | |      sec |-|   |
|         \2   12/ | |          \2/   |
|1 + --------------|*|1 - ------------|
|       2/x   5*pi\| |       2/x   pi\|
|    sec |- - ----|| |    sec |- - --||
\        \2    12 // \        \2   2 //
---------------------------------------
      /        2/x   pi\ \             
      |     sec |- + --| |             
      |         \2   12/ |    2/x\     
      |1 - --------------|*sec |-|     
      |       2/x   5*pi\|     \2/     
      |    sec |- - ----||             
      \        \2    12 //
(1+sec⁡2(x2+π12)sec⁡2(x2−5π12))(−sec⁡2(x2)sec⁡2(x2−π2)+1)(1−sec⁡2(x2+π12)sec⁡2(x2−5π12))sec⁡2(x2)\frac{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} + \frac{\pi}{12} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{5 \pi}{12} \right)}}\right) \left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)}{\left(1 - \frac{\sec^{2}{\left(\frac{x}{2} + \frac{\pi}{12} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{5 \pi}{12} \right)}}\right) \sec^{2}{\left(\frac{x}{2} \right)}} 
/       2/  x   5*pi\\ /       2/pi   x\\
|    csc |- - + ----|| |    csc |-- - -||
|        \  2    12 /| |        \2    2/|
|1 + ----------------|*|1 - ------------|
|         2/x   pi\  | |         2/x\   |
|      csc |- + --|  | |      csc |-|   |
\          \2   12/  / \          \2/   /
-----------------------------------------
   /       2/  x   5*pi\\                
   |    csc |- - + ----||                
   |        \  2    12 /|    2/pi   x\   
   |1 - ----------------|*csc |-- - -|   
   |         2/x   pi\  |     \2    2/   
   |      csc |- + --|  |                
   \          \2   12/  /
(1−csc⁡2(−x2+π2)csc⁡2(x2))(csc⁡2(−x2+5π12)csc⁡2(x2+π12)+1)(−csc⁡2(−x2+5π12)csc⁡2(x2+π12)+1)csc⁡2(−x2+π2)\frac{\left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \left(\frac{\csc^{2}{\left(- \frac{x}{2} + \frac{5 \pi}{12} \right)}}{\csc^{2}{\left(\frac{x}{2} + \frac{\pi}{12} \right)}} + 1\right)}{\left(- \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{5 \pi}{12} \right)}}{\csc^{2}{\left(\frac{x}{2} + \frac{\pi}{12} \right)}} + 1\right) \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} 
   2/x\ /       2/x   pi\\ /       2/x\\
cos |-|*|1 + tan |- + --||*|1 - tan |-||
    \2/ \        \2   12// \        \2//
----------------------------------------
                   2/x   pi\            
            1 - tan |- + --|            
                    \2   12/
(1−tan⁡2(x2))(tan⁡2(x2+π12)+1)cos⁡2(x2)1−tan⁡2(x2+π12)\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{12} \right)} + 1\right) \cos^{2}{\left(\frac{x}{2} \right)}}{1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{12} \right)}} 
        /         4/x\\ /         4/x   pi\\
        |    4*sin |-|| |    4*sin |- + --||
   2/x\ |          \2/| |          \2   12/|
cos |-|*|1 - ---------|*|1 + --------------|
    \2/ |        2    | |        2/    pi\ |
        \     sin (x) / |     sin |x + --| |
                        \         \    6 / /
--------------------------------------------
                      4/x   pi\             
                 4*sin |- + --|             
                       \2   12/             
             1 - --------------             
                     2/    pi\              
                  sin |x + --|              
                      \    6 /
(−4sin⁡4(x2)sin⁡2(x)+1)(4sin⁡4(x2+π12)sin⁡2(x+π6)+1)cos⁡2(x2)−4sin⁡4(x2+π12)sin⁡2(x+π6)+1\frac{\left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{12} \right)}}{\sin^{2}{\left(x + \frac{\pi}{6} \right)}} + 1\right) \cos^{2}{\left(\frac{x}{2} \right)}}{- \frac{4 \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{12} \right)}}{\sin^{2}{\left(x + \frac{\pi}{6} \right)}} + 1} 
/       2/x   pi\\ /        2/x\\
|1 + tan |- + --||*|-1 + cot |-||
\        \2   3 // \         \2//
---------------------------------
     /       2/x\\    /x   pi\   
   2*|1 + cot |-||*tan|- + --|   
     \        \2//    \2   3 /
(tan⁡2(x2+π3)+1)(cot⁡2(x2)−1)2(cot⁡2(x2)+1)tan⁡(x2+π3)\frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{3} \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{2 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \tan{\left(\frac{x}{2} + \frac{\pi}{3} \right)}} 
             2                                 
/       2/x\\  /       2/x   pi\\ /       2/x\\
|1 - tan |-|| *|1 + tan |- + --||*|1 - tan |-||
\        \4//  \        \2   12// \        \2//
-----------------------------------------------
                    2                          
       /       2/x\\  /       2/x   pi\\       
       |1 + tan |-|| *|1 - tan |- + --||       
       \        \4//  \        \2   12//
(1−tan⁡2(x4))2(1−tan⁡2(x2))(tan⁡2(x2+π12)+1)(1−tan⁡2(x2+π12))(tan⁡2(x4)+1)2\frac{\left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{2} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{12} \right)} + 1\right)}{\left(1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{12} \right)}\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}} 
             2                                         
/       2/x\\     4/x\ /       2/x   pi\\ /       2/x\\
|1 - tan |-|| *cos |-|*|1 + tan |- + --||*|1 - tan |-||
\        \4//      \4/ \        \2   12// \        \2//
-------------------------------------------------------
                           2/x   pi\                   
                    1 - tan |- + --|                   
                            \2   12/
(1−tan⁡2(x4))2(1−tan⁡2(x2))(tan⁡2(x2+π12)+1)cos⁡4(x4)1−tan⁡2(x2+π12)\frac{\left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{2} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{12} \right)} + 1\right) \cos^{4}{\left(\frac{x}{4} \right)}}{1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{12} \right)}} 
    /    pi\ 
 sin|x + --| 
    \    2 / 
-------------
   /    2*pi\
sin|x + ----|
   \     3  /
sin⁡(x+π2)sin⁡(x+2π3)\frac{\sin{\left(x + \frac{\pi}{2} \right)}}{\sin{\left(x + \frac{2 \pi}{3} \right)}} 
/       2/x   pi\\ /       2/x\\
|1 + tan |- + --||*|1 - tan |-||
\        \2   12// \        \2//
--------------------------------
/       2/x\\ /       2/x   pi\\
|1 + tan |-||*|1 - tan |- + --||
\        \2// \        \2   12//
(1−tan⁡2(x2))(tan⁡2(x2+π12)+1)(1−tan⁡2(x2+π12))(tan⁡2(x2)+1)\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{12} \right)} + 1\right)}{\left(1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{12} \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
(1 + tan(x/2 + pi/12)^2)*(1 - tan(x/2)^2)/((1 + tan(x/2)^2)*(1 - tan(x/2 + pi/12)^2))
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