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Descomponer al cuadrado:
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cos(10x)/(cos(5x)-sin(5x))
cos10x/cos5x-sin5x
cos(10*x) dividir por (cos(5*x)-sin(5*x))
Expresiones semejantes
cos(10*x)/(cos(5*x)+sin(5*x))
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cos(10*a)/(cos(5*a)-sin(5)*a)
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Simplificación de expresiones/ Descomposición en fracciones simples/ cos(10*x)/(cos(5*x)-sin(5*x))

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

Solución

Ha introducido [src]

     cos(10*x)     
-------------------
cos(5*x) - sin(5*x)

$$\frac{\cos{\left(10 x \right)}}{- \sin{\left(5 x \right)} + \cos{\left(5 x \right)}}$$

cos(10*x)/(cos(5*x) - sin(5*x))

Simplificación general [src]

  ___          
\/ 2 *cos(10*x)
---------------
     /      pi\
2*cos|5*x + --|
     \      4 /

$$\frac{\sqrt{2} \cos{\left(10 x \right)}}{2 \cos{\left(5 x + \frac{\pi}{4} \right)}}$$

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

$$\frac{\frac{e^{10 i x}}{2} + \frac{e^{- 10 i x}}{2}}{\frac{i \left(e^{5 i x} - e^{- 5 i x}\right)}{2} + \frac{e^{5 i x}}{2} + \frac{e^{- 5 i x}}{2}}$$

(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)

$$\frac{512 \cos^{10}{\left(x \right)}}{- 16 \sin^{5}{\left(x \right)} + 20 \sin^{3}{\left(x \right)} - 5 \sin{\left(x \right)} + 16 \cos^{5}{\left(x \right)} - 20 \cos^{3}{\left(x \right)} + 5 \cos{\left(x \right)}} - \frac{1280 \cos^{8}{\left(x \right)}}{- 16 \sin^{5}{\left(x \right)} + 20 \sin^{3}{\left(x \right)} - 5 \sin{\left(x \right)} + 16 \cos^{5}{\left(x \right)} - 20 \cos^{3}{\left(x \right)} + 5 \cos{\left(x \right)}} + \frac{1120 \cos^{6}{\left(x \right)}}{- 16 \sin^{5}{\left(x \right)} + 20 \sin^{3}{\left(x \right)} - 5 \sin{\left(x \right)} + 16 \cos^{5}{\left(x \right)} - 20 \cos^{3}{\left(x \right)} + 5 \cos{\left(x \right)}} - \frac{400 \cos^{4}{\left(x \right)}}{- 16 \sin^{5}{\left(x \right)} + 20 \sin^{3}{\left(x \right)} - 5 \sin{\left(x \right)} + 16 \cos^{5}{\left(x \right)} - 20 \cos^{3}{\left(x \right)} + 5 \cos{\left(x \right)}} + \frac{50 \cos^{2}{\left(x \right)}}{- 16 \sin^{5}{\left(x \right)} + 20 \sin^{3}{\left(x \right)} - 5 \sin{\left(x \right)} + 16 \cos^{5}{\left(x \right)} - 20 \cos^{3}{\left(x \right)} + 5 \cos{\left(x \right)}} - \frac{1}{- 16 \sin^{5}{\left(x \right)} + 20 \sin^{3}{\left(x \right)} - 5 \sin{\left(x \right)} + 16 \cos^{5}{\left(x \right)} - 20 \cos^{3}{\left(x \right)} + 5 \cos{\left(x \right)}}$$

-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       /

$$\frac{\sin{\left(10 x + \frac{\pi}{2} \right)}}{- \sin{\left(5 x \right)} + \sin{\left(5 x + \frac{\pi}{2} \right)}}$$

  ___              /       pi\
\/ 2 *cos(10*x)*csc|-5*x + --|
                   \       4 /
------------------------------
              2

$$\frac{\sqrt{2} \cos{\left(10 x \right)} \csc{\left(- 5 x + \frac{\pi}{4} \right)}}{2}$$

                        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 //

$$\frac{1 - \tan^{2}{\left(5 x \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{5 x}{2} \right)}}{\tan^{2}{\left(\frac{5 x}{2} \right)} + 1} - \frac{2 \tan{\left(\frac{5 x}{2} \right)}}{\tan^{2}{\left(\frac{5 x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(5 x \right)} + 1\right)}$$

                 1                  
------------------------------------
/   1             1      \          
|-------- - -------------|*sec(10*x)
|sec(5*x)      /      pi\|          
|           sec|5*x - --||          
\              \      2 //

$$\frac{1}{\left(- \frac{1}{\sec{\left(5 x - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(5 x \right)}}\right) \sec{\left(10 x \right)}}$$

  ___    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 //

$$\frac{\sqrt{2} \left(1 - \tan^{2}{\left(5 x \right)}\right) \left(\tan^{2}{\left(\frac{5 x}{2} + \frac{\pi}{8} \right)} + 1\right) \cos^{2}{\left(5 x \right)}}{2 \left(1 - \tan^{2}{\left(\frac{5 x}{2} + \frac{\pi}{8} \right)}\right)}$$

  ___ /       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 //

$$\frac{\sqrt{2} \left(1 - \tan^{2}{\left(5 x \right)}\right) \left(\tan^{2}{\left(\frac{5 x}{2} + \frac{\pi}{8} \right)} + 1\right)}{2 \left(1 - \tan^{2}{\left(\frac{5 x}{2} + \frac{\pi}{8} \right)}\right) \left(\tan^{2}{\left(5 x \right)} + 1\right)}$$

                         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 //

$$\frac{\cot^{2}{\left(5 x \right)} - 1}{\left(\frac{\cot^{2}{\left(\frac{5 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{5 x}{2} \right)} + 1} - \frac{2 \cot{\left(\frac{5 x}{2} \right)}}{\cot^{2}{\left(\frac{5 x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(5 x \right)} + 1\right)}$$

  ___    /      pi\
\/ 2 *sec|5*x + --|
         \      4 /
-------------------
    2*sec(10*x)

$$\frac{\sqrt{2} \sec{\left(5 x + \frac{\pi}{4} \right)}}{2 \sec{\left(10 x \right)}}$$

      ___    
    \/ 2     
-------------
   /      pi\
csc|5*x + --|
   \      4 /

$$\frac{\sqrt{2}}{\csc{\left(5 x + \frac{\pi}{4} \right)}}$$

  ___    /      pi\
\/ 2 *cos|5*x - --|
         \      4 /

$$\sqrt{2} \cos{\left(5 x - \frac{\pi}{4} \right)}$$

        cos(10*x)         
--------------------------
     /      pi\           
- cos|5*x - --| + cos(5*x)
     \      2 /

$$\frac{\cos{\left(10 x \right)}}{\cos{\left(5 x \right)} - \cos{\left(5 x - \frac{\pi}{2} \right)}}$$

               1               
-------------------------------
/   1          1    \          
|-------- - --------|*sec(10*x)
\sec(5*x)   csc(5*x)/

$$\frac{1}{\left(\frac{1}{\sec{\left(5 x \right)}} - \frac{1}{\csc{\left(5 x \right)}}\right) \sec{\left(10 x \right)}}$$

  ___              /      pi\
\/ 2 *cos(10*x)*sec|5*x + --|
                   \      4 /
-----------------------------
              2

$$\frac{\sqrt{2} \cos{\left(10 x \right)} \sec{\left(5 x + \frac{\pi}{4} \right)}}{2}$$

  ___    /       pi\
\/ 2 *csc|-5*x + --|
         \       4 /
--------------------
       /pi       \  
  2*csc|-- - 10*x|  
       \2        /

$$\frac{\sqrt{2} \csc{\left(- 5 x + \frac{\pi}{4} \right)}}{2 \csc{\left(- 10 x + \frac{\pi}{2} \right)}}$$

      ___    
    \/ 2     
-------------
   /      pi\
sec|5*x - --|
   \      4 /

$$\frac{\sqrt{2}}{\sec{\left(5 x - \frac{\pi}{4} \right)}}$$

  ___    /      pi\
\/ 2 *sin|5*x + --|
         \      4 /

$$\sqrt{2} \sin{\left(5 x + \frac{\pi}{4} \right)}$$

  ___    /pi       \
\/ 2 *sin|-- + 10*x|
         \2        /
--------------------
      /      3*pi\  
 2*sin|5*x + ----|  
      \       4  /

$$\frac{\sqrt{2} \sin{\left(10 x + \frac{\pi}{2} \right)}}{2 \sin{\left(5 x + \frac{3 \pi}{4} \right)}}$$

    ___    /pi   5*x\
2*\/ 2 *tan|-- + ---|
           \8     2 /
---------------------
         2/pi   5*x\ 
  1 + tan |-- + ---| 
          \8     2 /

$$\frac{2 \sqrt{2} \tan{\left(\frac{5 x}{2} + \frac{\pi}{8} \right)}}{\tan^{2}{\left(\frac{5 x}{2} + \frac{\pi}{8} \right)} + 1}$$

  ___ /       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 //

$$\frac{\sqrt{2} \left(\cot^{2}{\left(5 x \right)} - 1\right) \left(\cot^{2}{\left(\frac{5 x}{2} + \frac{\pi}{8} \right)} + 1\right)}{2 \left(\cot^{2}{\left(5 x \right)} + 1\right) \left(\cot^{2}{\left(\frac{5 x}{2} + \frac{\pi}{8} \right)} - 1\right)}$$

    ___    /pi   5*x\
2*\/ 2 *cot|-- + ---|
           \8     2 /
---------------------
         2/pi   5*x\ 
  1 + cot |-- + ---| 
          \8     2 /

$$\frac{2 \sqrt{2} \cot{\left(\frac{5 x}{2} + \frac{\pi}{8} \right)}}{\cot^{2}{\left(\frac{5 x}{2} + \frac{\pi}{8} \right)} + 1}$$

    ___    2/pi + 20*x\    /pi   5*x\
2*\/ 2 *sin |---------|*cot|-- + ---|
            \    8    /    \8     2 /

$$2 \sqrt{2} \sin^{2}{\left(\frac{20 x + \pi}{8} \right)} \cot{\left(\frac{5 x}{2} + \frac{\pi}{8} \right)}$$

  ___          
\/ 2 *cos(10*x)
---------------
     /      pi\
2*cos|5*x + --|
     \      4 /

$$\frac{\sqrt{2} \cos{\left(10 x \right)}}{2 \cos{\left(5 x + \frac{\pi}{4} \right)}}$$

                    1                    
-----------------------------------------
/      1            1    \    /pi       \
|------------- - --------|*csc|-- - 10*x|
|   /pi      \   csc(5*x)|    \2        /
|csc|-- - 5*x|           |               
\   \2       /           /

$$\frac{1}{\left(\frac{1}{\csc{\left(- 5 x + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(5 x \right)}}\right) \csc{\left(- 10 x + \frac{\pi}{2} \right)}}$$

                       ___                               
                    -\/ 2 *cos(10*x)                     
---------------------------------------------------------
                      ___________    ___________         
    ___              /       ___    /       ___          
- \/ 2 *cos(5*x) + \/  2 + \/ 2  *\/  2 - \/ 2  *sin(5*x)

$$- \frac{\sqrt{2} \cos{\left(10 x \right)}}{\sqrt{2 - \sqrt{2}} \sqrt{\sqrt{2} + 2} \sin{\left(5 x \right)} - \sqrt{2} \cos{\left(5 x \right)}}$$

-sqrt(2)*cos(10*x)/(-sqrt(2)*cos(5*x) + sqrt(2 + sqrt(2))*sqrt(2 - sqrt(2))*sin(5*x))

Otras calculadoras

¿Cómo usar?

Expresiones idénticas

Expresiones semejantes

Expresiones con funciones

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

Solución

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