Sr Examen

Otras calculadoras

Simplificación de expresiones/ Descomposición en fracciones simples/ cos(10*x)/(cos(5*x)-sin(5*x))

¿Cómo vas a descomponer esta cos(10*x)/(cos(5*x)-sin(5*x)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
     cos(10*x)     
-------------------
cos(5*x) - sin(5*x)
cos⁡(10x)−sin⁡(5x)+cos⁡(5x)\frac{\cos{\left(10 x \right)}}{- \sin{\left(5 x \right)} + \cos{\left(5 x \right)}} 
cos(10*x)/(cos(5*x) - sin(5*x))
Simplificación general [src] 
  ___          
\/ 2 *cos(10*x)
---------------
     /      pi\
2*cos|5*x + --|
     \      4 /
2cos⁡(10x)2cos⁡(5x+π4)\frac{\sqrt{2} \cos{\left(10 x \right)}}{2 \cos{\left(5 x + \frac{\pi}{4} \right)}} 
sqrt(2)*cos(10*x)/(2*cos(5*x + pi/4))
Respuesta numérica [src] 
cos(10*x)/(-sin(5*x) + cos(5*x))
cos(10*x)/(-sin(5*x) + cos(5*x))
Potencias [src] 
             -10*I*x    10*I*x           
            e          e                 
            -------- + -------           
               2          2              
-----------------------------------------
 -5*I*x    5*I*x     /   -5*I*x    5*I*x\
e         e        I*\- e       + e     /
------- + ------ + ----------------------
   2        2                2
e10ix2+e−10ix2i(e5ix−e−5ix)2+e5ix2+e−5ix2\frac{\frac{e^{10 i x}}{2} + \frac{e^{- 10 i x}}{2}}{\frac{i \left(e^{5 i x} - e^{- 5 i x}\right)}{2} + \frac{e^{5 i x}}{2} + \frac{e^{- 5 i x}}{2}} 
(exp(-10*i*x)/2 + exp(10*i*x)/2)/(exp(-5*i*x)/2 + exp(5*i*x)/2 + i*(-exp(-5*i*x) + exp(5*i*x))/2)
Abrimos la expresión [src] 
                                                                                                                     8                                                                          4                                                                           2                                                                           10                                                                           6                                 
                                      1                                                                      1280*cos (x)                                                                400*cos (x)                                                                  50*cos (x)                                                                 512*cos  (x)                                                                1120*cos (x)                              
- ------------------------------------------------------------------------- - ------------------------------------------------------------------------- - ------------------------------------------------------------------------- + ------------------------------------------------------------------------- + ------------------------------------------------------------------------- + -------------------------------------------------------------------------
          3            5                                  5            3              3            5                                  5            3              3            5                                  5            3              3            5                                  5            3              3            5                                  5            3              3            5                                  5            3   
  - 20*cos (x) - 16*sin (x) - 5*sin(x) + 5*cos(x) + 16*cos (x) + 20*sin (x)   - 20*cos (x) - 16*sin (x) - 5*sin(x) + 5*cos(x) + 16*cos (x) + 20*sin (x)   - 20*cos (x) - 16*sin (x) - 5*sin(x) + 5*cos(x) + 16*cos (x) + 20*sin (x)   - 20*cos (x) - 16*sin (x) - 5*sin(x) + 5*cos(x) + 16*cos (x) + 20*sin (x)   - 20*cos (x) - 16*sin (x) - 5*sin(x) + 5*cos(x) + 16*cos (x) + 20*sin (x)   - 20*cos (x) - 16*sin (x) - 5*sin(x) + 5*cos(x) + 16*cos (x) + 20*sin (x)
512cos⁡10(x)−16sin⁡5(x)+20sin⁡3(x)−5sin⁡(x)+16cos⁡5(x)−20cos⁡3(x)+5cos⁡(x)−1280cos⁡8(x)−16sin⁡5(x)+20sin⁡3(x)−5sin⁡(x)+16cos⁡5(x)−20cos⁡3(x)+5cos⁡(x)+1120cos⁡6(x)−16sin⁡5(x)+20sin⁡3(x)−5sin⁡(x)+16cos⁡5(x)−20cos⁡3(x)+5cos⁡(x)−400cos⁡4(x)−16sin⁡5(x)+20sin⁡3(x)−5sin⁡(x)+16cos⁡5(x)−20cos⁡3(x)+5cos⁡(x)+50cos⁡2(x)−16sin⁡5(x)+20sin⁡3(x)−5sin⁡(x)+16cos⁡5(x)−20cos⁡3(x)+5cos⁡(x)−1−16sin⁡5(x)+20sin⁡3(x)−5sin⁡(x)+16cos⁡5(x)−20cos⁡3(x)+5cos⁡(x)\frac{512 \cos^{10}{\left(x \right)}}{- 16 \sin^{5}{\left(x \right)} + 20 \sin^{3}{\left(x \right)} - 5 \sin{\left(x \right)} + 16 \cos^{5}{\left(x \right)} - 20 \cos^{3}{\left(x \right)} + 5 \cos{\left(x \right)}} - \frac{1280 \cos^{8}{\left(x \right)}}{- 16 \sin^{5}{\left(x \right)} + 20 \sin^{3}{\left(x \right)} - 5 \sin{\left(x \right)} + 16 \cos^{5}{\left(x \right)} - 20 \cos^{3}{\left(x \right)} + 5 \cos{\left(x \right)}} + \frac{1120 \cos^{6}{\left(x \right)}}{- 16 \sin^{5}{\left(x \right)} + 20 \sin^{3}{\left(x \right)} - 5 \sin{\left(x \right)} + 16 \cos^{5}{\left(x \right)} - 20 \cos^{3}{\left(x \right)} + 5 \cos{\left(x \right)}} - \frac{400 \cos^{4}{\left(x \right)}}{- 16 \sin^{5}{\left(x \right)} + 20 \sin^{3}{\left(x \right)} - 5 \sin{\left(x \right)} + 16 \cos^{5}{\left(x \right)} - 20 \cos^{3}{\left(x \right)} + 5 \cos{\left(x \right)}} + \frac{50 \cos^{2}{\left(x \right)}}{- 16 \sin^{5}{\left(x \right)} + 20 \sin^{3}{\left(x \right)} - 5 \sin{\left(x \right)} + 16 \cos^{5}{\left(x \right)} - 20 \cos^{3}{\left(x \right)} + 5 \cos{\left(x \right)}} - \frac{1}{- 16 \sin^{5}{\left(x \right)} + 20 \sin^{3}{\left(x \right)} - 5 \sin{\left(x \right)} + 16 \cos^{5}{\left(x \right)} - 20 \cos^{3}{\left(x \right)} + 5 \cos{\left(x \right)}} 
-1/(-20*cos(x)^3 - 16*sin(x)^5 - 5*sin(x) + 5*cos(x) + 16*cos(x)^5 + 20*sin(x)^3) - 1280*cos(x)^8/(-20*cos(x)^3 - 16*sin(x)^5 - 5*sin(x) + 5*cos(x) + 16*cos(x)^5 + 20*sin(x)^3) - 400*cos(x)^4/(-20*cos(x)^3 - 16*sin(x)^5 - 5*sin(x) + 5*cos(x) + 16*cos(x)^5 + 20*sin(x)^3) + 50*cos(x)^2/(-20*cos(x)^3 - 16*sin(x)^5 - 5*sin(x) + 5*cos(x) + 16*cos(x)^5 + 20*sin(x)^3) + 512*cos(x)^10/(-20*cos(x)^3 - 16*sin(x)^5 - 5*sin(x) + 5*cos(x) + 16*cos(x)^5 + 20*sin(x)^3) + 1120*cos(x)^6/(-20*cos(x)^3 - 16*sin(x)^5 - 5*sin(x) + 5*cos(x) + 16*cos(x)^5 + 20*sin(x)^3)
Parte trigonométrica [src] 
         /pi       \     
      sin|-- + 10*x|     
         \2        /     
-------------------------
               /pi      \
-sin(5*x) + sin|-- + 5*x|
               \2       /
sin⁡(10x+π2)−sin⁡(5x)+sin⁡(5x+π2)\frac{\sin{\left(10 x + \frac{\pi}{2} \right)}}{- \sin{\left(5 x \right)} + \sin{\left(5 x + \frac{\pi}{2} \right)}} 
  ___              /       pi\
\/ 2 *cos(10*x)*csc|-5*x + --|
                   \       4 /
------------------------------
              2
2cos⁡(10x)csc⁡(−5x+π4)2\frac{\sqrt{2} \cos{\left(10 x \right)} \csc{\left(- 5 x + \frac{\pi}{4} \right)}}{2} 
                        2                      
                 1 - tan (5*x)                 
-----------------------------------------------
                /       2/5*x\          /5*x\ \
                |1 - tan |---|     2*tan|---| |
/       2     \ |        \ 2 /          \ 2 / |
\1 + tan (5*x)/*|------------- - -------------|
                |       2/5*x\          2/5*x\|
                |1 + tan |---|   1 + tan |---||
                \        \ 2 /           \ 2 //
1−tan⁡2(5x)(1−tan⁡2(5x2)tan⁡2(5x2)+1−2tan⁡(5x2)tan⁡2(5x2)+1)(tan⁡2(5x)+1)\frac{1 - \tan^{2}{\left(5 x \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{5 x}{2} \right)}}{\tan^{2}{\left(\frac{5 x}{2} \right)} + 1} - \frac{2 \tan{\left(\frac{5 x}{2} \right)}}{\tan^{2}{\left(\frac{5 x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(5 x \right)} + 1\right)} 
                 1                  
------------------------------------
/   1             1      \          
|-------- - -------------|*sec(10*x)
|sec(5*x)      /      pi\|          
|           sec|5*x - --||          
\              \      2 //
1(−1sec⁡(5x−π2)+1sec⁡(5x))sec⁡(10x)\frac{1}{\left(- \frac{1}{\sec{\left(5 x - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(5 x \right)}}\right) \sec{\left(10 x \right)}} 
  ___    2      /       2/pi   5*x\\ /       2     \
\/ 2 *cos (5*x)*|1 + tan |-- + ---||*\1 - tan (5*x)/
                \        \8     2 //                
----------------------------------------------------
                 /       2/pi   5*x\\               
               2*|1 - tan |-- + ---||               
                 \        \8     2 //
2(1−tan⁡2(5x))(tan⁡2(5x2+π8)+1)cos⁡2(5x)2(1−tan⁡2(5x2+π8))\frac{\sqrt{2} \left(1 - \tan^{2}{\left(5 x \right)}\right) \left(\tan^{2}{\left(\frac{5 x}{2} + \frac{\pi}{8} \right)} + 1\right) \cos^{2}{\left(5 x \right)}}{2 \left(1 - \tan^{2}{\left(\frac{5 x}{2} + \frac{\pi}{8} \right)}\right)} 
  ___ /       2/pi   5*x\\ /       2     \
\/ 2 *|1 + tan |-- + ---||*\1 - tan (5*x)/
      \        \8     2 //                
------------------------------------------
    /       2     \ /       2/pi   5*x\\  
  2*\1 + tan (5*x)/*|1 - tan |-- + ---||  
                    \        \8     2 //
2(1−tan⁡2(5x))(tan⁡2(5x2+π8)+1)2(1−tan⁡2(5x2+π8))(tan⁡2(5x)+1)\frac{\sqrt{2} \left(1 - \tan^{2}{\left(5 x \right)}\right) \left(\tan^{2}{\left(\frac{5 x}{2} + \frac{\pi}{8} \right)} + 1\right)}{2 \left(1 - \tan^{2}{\left(\frac{5 x}{2} + \frac{\pi}{8} \right)}\right) \left(\tan^{2}{\left(5 x \right)} + 1\right)} 
                         2                      
                 -1 + cot (5*x)                 
------------------------------------------------
                /        2/5*x\          /5*x\ \
                |-1 + cot |---|     2*cot|---| |
/       2     \ |         \ 2 /          \ 2 / |
\1 + cot (5*x)/*|-------------- - -------------|
                |       2/5*x\           2/5*x\|
                |1 + cot |---|    1 + cot |---||
                \        \ 2 /            \ 2 //
cot⁡2(5x)−1(cot⁡2(5x2)−1cot⁡2(5x2)+1−2cot⁡(5x2)cot⁡2(5x2)+1)(cot⁡2(5x)+1)\frac{\cot^{2}{\left(5 x \right)} - 1}{\left(\frac{\cot^{2}{\left(\frac{5 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{5 x}{2} \right)} + 1} - \frac{2 \cot{\left(\frac{5 x}{2} \right)}}{\cot^{2}{\left(\frac{5 x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(5 x \right)} + 1\right)} 
  ___    /      pi\
\/ 2 *sec|5*x + --|
         \      4 /
-------------------
    2*sec(10*x)
2sec⁡(5x+π4)2sec⁡(10x)\frac{\sqrt{2} \sec{\left(5 x + \frac{\pi}{4} \right)}}{2 \sec{\left(10 x \right)}} 
      ___    
    \/ 2     
-------------
   /      pi\
csc|5*x + --|
   \      4 /
2csc⁡(5x+π4)\frac{\sqrt{2}}{\csc{\left(5 x + \frac{\pi}{4} \right)}} 
  ___    /      pi\
\/ 2 *cos|5*x - --|
         \      4 /
2cos⁡(5x−π4)\sqrt{2} \cos{\left(5 x - \frac{\pi}{4} \right)} 
        cos(10*x)         
--------------------------
     /      pi\           
- cos|5*x - --| + cos(5*x)
     \      2 /
cos⁡(10x)cos⁡(5x)−cos⁡(5x−π2)\frac{\cos{\left(10 x \right)}}{\cos{\left(5 x \right)} - \cos{\left(5 x - \frac{\pi}{2} \right)}} 
               1               
-------------------------------
/   1          1    \          
|-------- - --------|*sec(10*x)
\sec(5*x)   csc(5*x)/
1(1sec⁡(5x)−1csc⁡(5x))sec⁡(10x)\frac{1}{\left(\frac{1}{\sec{\left(5 x \right)}} - \frac{1}{\csc{\left(5 x \right)}}\right) \sec{\left(10 x \right)}} 
  ___              /      pi\
\/ 2 *cos(10*x)*sec|5*x + --|
                   \      4 /
-----------------------------
              2
2cos⁡(10x)sec⁡(5x+π4)2\frac{\sqrt{2} \cos{\left(10 x \right)} \sec{\left(5 x + \frac{\pi}{4} \right)}}{2} 
  ___    /       pi\
\/ 2 *csc|-5*x + --|
         \       4 /
--------------------
       /pi       \  
  2*csc|-- - 10*x|  
       \2        /
2csc⁡(−5x+π4)2csc⁡(−10x+π2)\frac{\sqrt{2} \csc{\left(- 5 x + \frac{\pi}{4} \right)}}{2 \csc{\left(- 10 x + \frac{\pi}{2} \right)}} 
      ___    
    \/ 2     
-------------
   /      pi\
sec|5*x - --|
   \      4 /
2sec⁡(5x−π4)\frac{\sqrt{2}}{\sec{\left(5 x - \frac{\pi}{4} \right)}} 
  ___    /      pi\
\/ 2 *sin|5*x + --|
         \      4 /
2sin⁡(5x+π4)\sqrt{2} \sin{\left(5 x + \frac{\pi}{4} \right)} 
  ___    /pi       \
\/ 2 *sin|-- + 10*x|
         \2        /
--------------------
      /      3*pi\  
 2*sin|5*x + ----|  
      \       4  /
2sin⁡(10x+π2)2sin⁡(5x+3π4)\frac{\sqrt{2} \sin{\left(10 x + \frac{\pi}{2} \right)}}{2 \sin{\left(5 x + \frac{3 \pi}{4} \right)}} 
    ___    /pi   5*x\
2*\/ 2 *tan|-- + ---|
           \8     2 /
---------------------
         2/pi   5*x\ 
  1 + tan |-- + ---| 
          \8     2 /
22tan⁡(5x2+π8)tan⁡2(5x2+π8)+1\frac{2 \sqrt{2} \tan{\left(\frac{5 x}{2} + \frac{\pi}{8} \right)}}{\tan^{2}{\left(\frac{5 x}{2} + \frac{\pi}{8} \right)} + 1} 
  ___ /       2/pi   5*x\\ /        2     \
\/ 2 *|1 + cot |-- + ---||*\-1 + cot (5*x)/
      \        \8     2 //                 
-------------------------------------------
    /       2     \ /        2/pi   5*x\\  
  2*\1 + cot (5*x)/*|-1 + cot |-- + ---||  
                    \         \8     2 //
2(cot⁡2(5x)−1)(cot⁡2(5x2+π8)+1)2(cot⁡2(5x)+1)(cot⁡2(5x2+π8)−1)\frac{\sqrt{2} \left(\cot^{2}{\left(5 x \right)} - 1\right) \left(\cot^{2}{\left(\frac{5 x}{2} + \frac{\pi}{8} \right)} + 1\right)}{2 \left(\cot^{2}{\left(5 x \right)} + 1\right) \left(\cot^{2}{\left(\frac{5 x}{2} + \frac{\pi}{8} \right)} - 1\right)} 
    ___    /pi   5*x\
2*\/ 2 *cot|-- + ---|
           \8     2 /
---------------------
         2/pi   5*x\ 
  1 + cot |-- + ---| 
          \8     2 /
22cot⁡(5x2+π8)cot⁡2(5x2+π8)+1\frac{2 \sqrt{2} \cot{\left(\frac{5 x}{2} + \frac{\pi}{8} \right)}}{\cot^{2}{\left(\frac{5 x}{2} + \frac{\pi}{8} \right)} + 1} 
    ___    2/pi + 20*x\    /pi   5*x\
2*\/ 2 *sin |---------|*cot|-- + ---|
            \    8    /    \8     2 /
22sin⁡2(20x+π8)cot⁡(5x2+π8)2 \sqrt{2} \sin^{2}{\left(\frac{20 x + \pi}{8} \right)} \cot{\left(\frac{5 x}{2} + \frac{\pi}{8} \right)} 
  ___          
\/ 2 *cos(10*x)
---------------
     /      pi\
2*cos|5*x + --|
     \      4 /
2cos⁡(10x)2cos⁡(5x+π4)\frac{\sqrt{2} \cos{\left(10 x \right)}}{2 \cos{\left(5 x + \frac{\pi}{4} \right)}} 
                    1                    
-----------------------------------------
/      1            1    \    /pi       \
|------------- - --------|*csc|-- - 10*x|
|   /pi      \   csc(5*x)|    \2        /
|csc|-- - 5*x|           |               
\   \2       /           /
1(1csc⁡(−5x+π2)−1csc⁡(5x))csc⁡(−10x+π2)\frac{1}{\left(\frac{1}{\csc{\left(- 5 x + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(5 x \right)}}\right) \csc{\left(- 10 x + \frac{\pi}{2} \right)}} 
                       ___                               
                    -\/ 2 *cos(10*x)                     
---------------------------------------------------------
                      ___________    ___________         
    ___              /       ___    /       ___          
- \/ 2 *cos(5*x) + \/  2 + \/ 2  *\/  2 - \/ 2  *sin(5*x)
−2cos⁡(10x)2−22+2sin⁡(5x)−2cos⁡(5x)- \frac{\sqrt{2} \cos{\left(10 x \right)}}{\sqrt{2 - \sqrt{2}} \sqrt{\sqrt{2} + 2} \sin{\left(5 x \right)} - \sqrt{2} \cos{\left(5 x \right)}} 
-sqrt(2)*cos(10*x)/(-sqrt(2)*cos(5*x) + sqrt(2 + sqrt(2))*sqrt(2 - sqrt(2))*sin(5*x))
    © Sr Examen – Калькулятор