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Simplificación de expresiones/ Descomposición en fracciones simples/ sin^2(x)-cos^2(x)+cos^4(x)/(cos^2(x)-sin^2(x)+sin^4(x))

¿Cómo vas a descomponer esta sin^2(x)-cos^2(x)+cos^4(x)/(cos^2(x)-sin^2(x)+sin^4(x)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
                                 4             
   2         2                cos (x)          
sin (x) - cos (x) + ---------------------------
                       2         2         4   
                    cos (x) - sin (x) + sin (x)
(sin⁡2(x)−cos⁡2(x))+cos⁡4(x)(−sin⁡2(x)+cos⁡2(x))+sin⁡4(x)\left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) + \frac{\cos^{4}{\left(x \right)}}{\left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) + \sin^{4}{\left(x \right)}} 
sin(x)^2 - cos(x)^2 + cos(x)^4/(cos(x)^2 - sin(x)^2 + sin(x)^4)
Simplificación general [src] 
     2   
2*sin (x)
2sin⁡2(x)2 \sin^{2}{\left(x \right)} 
2*sin(x)^2
Potencias [src] 
                                                                          4                    
                                                            / I*x    -I*x\                     
                2                   2                       |e      e    |                     
  / I*x    -I*x\    /   -I*x    I*x\                        |---- + -----|                     
  |e      e    |    \- e     + e   /                        \ 2       2  /                     
- |---- + -----|  - ----------------- + -------------------------------------------------------
  \ 2       2  /            4                         2                   2                   4
                                        / I*x    -I*x\    /   -I*x    I*x\    /   -I*x    I*x\ 
                                        |e      e    |    \- e     + e   /    \- e     + e   / 
                                        |---- + -----|  + ----------------- + -----------------
                                        \ 2       2  /            4                   16
(eix2+e−ix2)4(eix2+e−ix2)2+(eix−e−ix)416+(eix−e−ix)24−(eix2+e−ix2)2−(eix−e−ix)24\frac{\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{4}}{\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2} + \frac{\left(e^{i x} - e^{- i x}\right)^{4}}{16} + \frac{\left(e^{i x} - e^{- i x}\right)^{2}}{4}} - \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2} - \frac{\left(e^{i x} - e^{- i x}\right)^{2}}{4} 
                                 4             
   2         2                cos (x)          
sin (x) - cos (x) + ---------------------------
                       2         4         2   
                    cos (x) + sin (x) - sin (x)
sin⁡2(x)−cos⁡2(x)+cos⁡4(x)sin⁡4(x)−sin⁡2(x)+cos⁡2(x)\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} + \frac{\cos^{4}{\left(x \right)}}{\sin^{4}{\left(x \right)} - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}} 
sin(x)^2 - cos(x)^2 + cos(x)^4/(cos(x)^2 + sin(x)^4 - sin(x)^2)
Respuesta numérica [src] 
sin(x)^2 - cos(x)^2 + cos(x)^4/(cos(x)^2 + sin(x)^4 - sin(x)^2)
sin(x)^2 - cos(x)^2 + cos(x)^4/(cos(x)^2 + sin(x)^4 - sin(x)^2)
Denominador racional [src] 
   4      /   2         2   \ /   2         4         2   \
cos (x) + \sin (x) - cos (x)/*\cos (x) + sin (x) - sin (x)/
-----------------------------------------------------------
                   2         4         2                   
                cos (x) + sin (x) - sin (x)
(sin⁡2(x)−cos⁡2(x))(sin⁡4(x)−sin⁡2(x)+cos⁡2(x))+cos⁡4(x)sin⁡4(x)−sin⁡2(x)+cos⁡2(x)\frac{\left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \left(\sin^{4}{\left(x \right)} - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) + \cos^{4}{\left(x \right)}}{\sin^{4}{\left(x \right)} - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}} 
(cos(x)^4 + (sin(x)^2 - cos(x)^2)*(cos(x)^2 + sin(x)^4 - sin(x)^2))/(cos(x)^2 + sin(x)^4 - sin(x)^2)
Combinatoria [src] 
    2    /   2         4           2         2       2   \ 
-sin (x)*\sin (x) - sin (x) - 2*cos (x) + cos (x)*sin (x)/ 
-----------------------------------------------------------
                   2         4         2                   
                cos (x) + sin (x) - sin (x)
−(−sin⁡4(x)+sin⁡2(x)cos⁡2(x)+sin⁡2(x)−2cos⁡2(x))sin⁡2(x)sin⁡4(x)−sin⁡2(x)+cos⁡2(x)- \frac{\left(- \sin^{4}{\left(x \right)} + \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} + \sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)}}{\sin^{4}{\left(x \right)} - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}} 
-sin(x)^2*(sin(x)^2 - sin(x)^4 - 2*cos(x)^2 + cos(x)^2*sin(x)^2)/(cos(x)^2 + sin(x)^4 - sin(x)^2)
Unión de expresiones racionales [src] 
   4      /   2         2   \ /   2         4         2   \
cos (x) + \sin (x) - cos (x)/*\cos (x) + sin (x) - sin (x)/
-----------------------------------------------------------
                   2         4         2                   
                cos (x) + sin (x) - sin (x)
(sin⁡2(x)−cos⁡2(x))(sin⁡4(x)−sin⁡2(x)+cos⁡2(x))+cos⁡4(x)sin⁡4(x)−sin⁡2(x)+cos⁡2(x)\frac{\left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \left(\sin^{4}{\left(x \right)} - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) + \cos^{4}{\left(x \right)}}{\sin^{4}{\left(x \right)} - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}} 
(cos(x)^4 + (sin(x)^2 - cos(x)^2)*(cos(x)^2 + sin(x)^4 - sin(x)^2))/(cos(x)^2 + sin(x)^4 - sin(x)^2)
Compilar la expresión [src] 
                                 4             
   2         2                cos (x)          
sin (x) - cos (x) + ---------------------------
                       2         4         2   
                    cos (x) + sin (x) - sin (x)
sin⁡2(x)−cos⁡2(x)+cos⁡4(x)sin⁡4(x)−sin⁡2(x)+cos⁡2(x)\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} + \frac{\cos^{4}{\left(x \right)}}{\sin^{4}{\left(x \right)} - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}} 
sin(x)^2 - cos(x)^2 + cos(x)^4/(cos(x)^2 + sin(x)^4 - sin(x)^2)
Parte trigonométrica [src] 
               2                                                           4                         
  /       2/x\\           2/x\                                /       2/x\\                          
  |1 - tan |-||      4*tan |-|                                |1 - tan |-||                          
  \        \2//            \2/                                \        \2//                          
- -------------- + -------------- + -----------------------------------------------------------------
               2                2                  /             2                                  \
  /       2/x\\    /       2/x\\                   |/       2/x\\           2/x\              4/x\  |
  |1 + tan |-||    |1 + tan |-||                 4 ||1 - tan |-||      4*tan |-|        16*tan |-|  |
  \        \2//    \        \2//    /       2/x\\  |\        \2//            \2/               \2/  |
                                    |1 + tan |-|| *|-------------- - -------------- + --------------|
                                    \        \2//  |             2                2                4|
                                                   |/       2/x\\    /       2/x\\    /       2/x\\ |
                                                   ||1 + tan |-||    |1 + tan |-||    |1 + tan |-|| |
                                                   \\        \2//    \        \2//    \        \2// /
(1−tan⁡2(x2))4(tan⁡2(x2)+1)4((1−tan⁡2(x2))2(tan⁡2(x2)+1)2−4tan⁡2(x2)(tan⁡2(x2)+1)2+16tan⁡4(x2)(tan⁡2(x2)+1)4)−(1−tan⁡2(x2))2(tan⁡2(x2)+1)2+4tan⁡2(x2)(tan⁡2(x2)+1)2\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{4}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4} \left(\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} - \frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{16 \tan^{4}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}}\right)} - \frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} 
   1         1                        1                  
------- - ------- + -------------------------------------
   2         2      /   1         1         1   \    4   
csc (x)   sec (x)   |------- + ------- - -------|*sec (x)
                    |   4         2         2   |        
                    \csc (x)   sec (x)   csc (x)/
−1sec⁡2(x)+1csc⁡2(x)+1(1sec⁡2(x)−1csc⁡2(x)+1csc⁡4(x))sec⁡4(x)- \frac{1}{\sec^{2}{\left(x \right)}} + \frac{1}{\csc^{2}{\left(x \right)}} + \frac{1}{\left(\frac{1}{\sec^{2}{\left(x \right)}} - \frac{1}{\csc^{2}{\left(x \right)}} + \frac{1}{\csc^{4}{\left(x \right)}}\right) \sec^{4}{\left(x \right)}} 
                                 4               4        
                    /       2/x\\  /        1   \     8/x\
                256*|1 + tan |-|| *|-1 + -------| *tan |-|
                    \        \2//  |        2/x\|      \4/
         2                         |     tan |-||         
  1 - tan (x)                      \         \2//         
- ----------- + ------------------------------------------
         2                         8              4       
  1 + tan (x)         /       2/x\\  /       2/x\\        
                      |1 + tan |-|| *|1 - tan |-||        
                      \        \4//  \        \2//
256(−1+1tan⁡2(x2))4(tan⁡2(x2)+1)4tan⁡8(x4)(1−tan⁡2(x2))4(tan⁡2(x4)+1)8−1−tan⁡2(x)tan⁡2(x)+1\frac{256 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4} \tan^{8}{\left(\frac{x}{4} \right)}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{4} \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} - \frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} 
                              4        
            /       2/x   pi\\         
            |    cos |- - --||         
            |        \2   2 /|     8/x\
            |1 - ------------| *cos |-|
            |         2/x\   |      \2/
            |      cos |-|   |         
            \          \2/   /         
-cos(2*x) + ---------------------------
                         4             
                      cos (x)
(1−cos⁡2(x2−π2)cos⁡2(x2))4cos⁡8(x2)cos⁡4(x)−cos⁡(2x)\frac{\left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \cos^{8}{\left(\frac{x}{2} \right)}}{\cos^{4}{\left(x \right)}} - \cos{\left(2 x \right)} 
                         4                
            /       2/x\\     8/x\    4   
-cos(2*x) + |1 - tan |-|| *cos |-|*sec (x)
            \        \2//      \2/
(1−tan⁡2(x2))4cos⁡8(x2)sec⁡4(x)−cos⁡(2x)\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{4} \cos^{8}{\left(\frac{x}{2} \right)} \sec^{4}{\left(x \right)} - \cos{\left(2 x \right)} 
                            4        
            /         2    \         
            |      sin (x) |     8/x\
            |-1 + ---------| *sin |-|
            |          4/x\|      \2/
            |     4*sin |-||         
            \           \2//         
-cos(2*x) + -------------------------
                        4            
                     cos (x)
(−1+sin⁡2(x)4sin⁡4(x2))4sin⁡8(x2)cos⁡4(x)−cos⁡(2x)\frac{\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)}}{\cos^{4}{\left(x \right)}} - \cos{\left(2 x \right)} 
                                  4        
                     /       2/x\\     8/x\
          2      256*|1 + cot |-|| *cot |-|
  -1 + cot (x)       \        \2//      \4/
- ------------ + --------------------------
         2                          8      
  1 + cot (x)          /       2/x\\       
                       |1 + cot |-||       
                       \        \4//
−cot⁡2(x)−1cot⁡2(x)+1+256(cot⁡2(x2)+1)4cot⁡8(x4)(cot⁡2(x4)+1)8- \frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} + \frac{256 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4} \cot^{8}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} 
1 - cos(2*x)
1−cos⁡(2x)1 - \cos{\left(2 x \right)} 
   2   
-------
   2   
csc (x)
2csc⁡2(x)\frac{2}{\csc^{2}{\left(x \right)}} 
                                      4/    pi\          
                                   sin |x + --|          
   2         2/    pi\                 \    2 /          
sin (x) - sin |x + --| + --------------------------------
              \    2 /      4         2/    pi\      2   
                         sin (x) + sin |x + --| - sin (x)
                                       \    2 /
sin⁡2(x)−sin⁡2(x+π2)+sin⁡4(x+π2)sin⁡4(x)−sin⁡2(x)+sin⁡2(x+π2)\sin^{2}{\left(x \right)} - \sin^{2}{\left(x + \frac{\pi}{2} \right)} + \frac{\sin^{4}{\left(x + \frac{\pi}{2} \right)}}{\sin^{4}{\left(x \right)} - \sin^{2}{\left(x \right)} + \sin^{2}{\left(x + \frac{\pi}{2} \right)}} 
                                    4             
                  /       2/pi   x\\              
                  |    csc |-- - -||              
                  |        \2    2/|     4/pi    \
                  |1 - ------------| *csc |-- - x|
                  |         2/x\   |      \2     /
                  |      csc |-|   |              
        1         \          \2/   /              
- ------------- + --------------------------------
     /pi      \                8/pi   x\          
  csc|-- - 2*x|             csc |-- - -|          
     \2       /                 \2    2/
(1−csc⁡2(−x2+π2)csc⁡2(x2))4csc⁡4(−x+π2)csc⁡8(−x2+π2)−1csc⁡(−2x+π2)\frac{\left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \csc^{4}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{8}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}} 
                                 4             
                  /         4/x\\              
                  |    4*sin |-||              
                  |          \2/|     8/pi   x\
                  |1 - ---------| *sin |-- + -|
                  |        2    |      \2    2/
     /pi      \   \     sin (x) /              
- sin|-- + 2*x| + -----------------------------
     \2       /               4/    pi\        
                           sin |x + --|        
                               \    2 /
(−4sin⁡4(x2)sin⁡2(x)+1)4sin⁡8(x2+π2)sin⁡4(x+π2)−sin⁡(2x+π2)\frac{\left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{4} \sin^{8}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{\sin^{4}{\left(x + \frac{\pi}{2} \right)}} - \sin{\left(2 x + \frac{\pi}{2} \right)} 
       2/x\   
  8*cot |-|   
        \2/   
--------------
             2
/       2/x\\ 
|1 + cot |-|| 
\        \2//
8cot⁡2(x2)(cot⁡2(x2)+1)2\frac{8 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} 
                2                                                           4                          
  /        2/x\\           2/x\                               /        2/x\\                           
  |-1 + cot |-||      4*cot |-|                               |-1 + cot |-||                           
  \         \2//            \2/                               \         \2//                           
- --------------- + -------------- + ------------------------------------------------------------------
                2                2                  /              2                                  \
   /       2/x\\    /       2/x\\                   |/        2/x\\           2/x\              4/x\  |
   |1 + cot |-||    |1 + cot |-||                 4 ||-1 + cot |-||      4*cot |-|        16*cot |-|  |
   \        \2//    \        \2//    /       2/x\\  |\         \2//            \2/               \2/  |
                                     |1 + cot |-|| *|--------------- - -------------- + --------------|
                                     \        \2//  |              2                2                4|
                                                    | /       2/x\\    /       2/x\\    /       2/x\\ |
                                                    | |1 + cot |-||    |1 + cot |-||    |1 + cot |-|| |
                                                    \ \        \2//    \        \2//    \        \2// /
(cot⁡2(x2)−1)4(cot⁡2(x2)+1)4((cot⁡2(x2)−1)2(cot⁡2(x2)+1)2−4cot⁡2(x2)(cot⁡2(x2)+1)2+16cot⁡4(x2)(cot⁡2(x2)+1)4)−(cot⁡2(x2)−1)2(cot⁡2(x2)+1)2+4cot⁡2(x2)(cot⁡2(x2)+1)2\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4} \left(\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} - \frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{16 \cot^{4}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}}\right)} - \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} 
                              4              4               8
                 /       2/x\\  /       1   \  /        2/x\\ 
                 |1 + cot |-|| *|1 - -------| *|-1 + cot |-|| 
                 \        \2//  |       2/x\|  \         \4// 
          2                     |    cot |-||                 
  -1 + cot (x)                  \        \2//                 
- ------------ + ---------------------------------------------
         2                            8               4       
  1 + cot (x)            /       2/x\\  /        2/x\\        
                         |1 + cot |-|| *|-1 + cot |-||        
                         \        \4//  \         \2//
(1−1cot⁡2(x2))4(cot⁡2(x4)−1)8(cot⁡2(x2)+1)4(cot⁡2(x4)+1)8(cot⁡2(x2)−1)4−cot⁡2(x)−1cot⁡2(x)+1\frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{8} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4}} - \frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} 
                        8              8        
            /       /x\\  /        /x\\     8/x\
            |1 + tan|-|| *|-1 + tan|-|| *tan |-|
            \       \4//  \        \4//      \2/
-cos(2*x) + ------------------------------------
                               8/x\             
                        256*tan |-|             
                                \4/
(tan⁡(x4)−1)8(tan⁡(x4)+1)8tan⁡8(x2)256tan⁡8(x4)−cos⁡(2x)\frac{\left(\tan{\left(\frac{x}{4} \right)} - 1\right)^{8} \left(\tan{\left(\frac{x}{4} \right)} + 1\right)^{8} \tan^{8}{\left(\frac{x}{2} \right)}}{256 \tan^{8}{\left(\frac{x}{4} \right)}} - \cos{\left(2 x \right)} 
       2/x\   
  8*tan |-|   
        \2/   
--------------
             2
/       2/x\\ 
|1 + tan |-|| 
\        \2//
8tan⁡2(x2)(tan⁡2(x2)+1)2\frac{8 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} 
   1           1                                1                       
------- - ------------ + -----------------------------------------------
   2         2/pi    \   /   1           1            1   \    4/pi    \
csc (x)   csc |-- - x|   |------- + ------------ - -------|*csc |-- - x|
              \2     /   |   4         2/pi    \      2   |     \2     /
                         |csc (x)   csc |-- - x|   csc (x)|             
                         \              \2     /          /
−1csc⁡2(−x+π2)+1csc⁡2(x)+1(1csc⁡2(−x+π2)−1csc⁡2(x)+1csc⁡4(x))csc⁡4(−x+π2)- \frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{2}{\left(x \right)}} + \frac{1}{\left(\frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} - \frac{1}{\csc^{2}{\left(x \right)}} + \frac{1}{\csc^{4}{\left(x \right)}}\right) \csc^{4}{\left(- x + \frac{\pi}{2} \right)}} 
                          4        
            /        1   \     8/x\
            |-1 + -------| *sin |-|
            |        2/x\|      \2/
            |     tan |-||         
            \         \2//         
-cos(2*x) + -----------------------
                       4           
                    cos (x)
(−1+1tan⁡2(x2))4sin⁡8(x2)cos⁡4(x)−cos⁡(2x)\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)}}{\cos^{4}{\left(x \right)}} - \cos{\left(2 x \right)} 
     1            1                             1                       
------------ - ------- + -----------------------------------------------
   2/    pi\      2      /   1           1              1      \    4   
sec |x - --|   sec (x)   |------- + ------------ - ------------|*sec (x)
    \    2 /             |   2         4/    pi\      2/    pi\|        
                         |sec (x)   sec |x - --|   sec |x - --||        
                         \              \    2 /       \    2 //
1sec⁡2(x−π2)−1sec⁡2(x)+1(−1sec⁡2(x−π2)+1sec⁡4(x−π2)+1sec⁡2(x))sec⁡4(x)\frac{1}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} - \frac{1}{\sec^{2}{\left(x \right)}} + \frac{1}{\left(- \frac{1}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec^{4}{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec^{2}{\left(x \right)}}\right) \sec^{4}{\left(x \right)}} 
                                            4        
                              /        2/x\\     8/x\
                              |-1 + cot |-|| *sin |-|
1 - cos(2*x)   1 + cos(2*x)   \         \2//      \2/
------------ - ------------ + -----------------------
     2              2                    4           
                                      cos (x)
1−cos⁡(2x)2−cos⁡(2x)+12+(cot⁡2(x2)−1)4sin⁡8(x2)cos⁡4(x)\frac{1 - \cos{\left(2 x \right)}}{2} - \frac{\cos{\left(2 x \right)} + 1}{2} + \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)}}{\cos^{4}{\left(x \right)}} 
                                  4        
                  /         2    \         
                  |      sin (x) |     8/x\
                  |-1 + ---------| *sin |-|
                  |          4/x\|      \2/
                  |     4*sin |-||         
     /pi      \   \           \2//         
- sin|-- + 2*x| + -------------------------
     \2       /             4/    pi\      
                         sin |x + --|      
                             \    2 /
(−1+sin⁡2(x)4sin⁡4(x2))4sin⁡8(x2)sin⁡4(x+π2)−sin⁡(2x+π2)\frac{\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)}}{\sin^{4}{\left(x + \frac{\pi}{2} \right)}} - \sin{\left(2 x + \frac{\pi}{2} \right)} 
     2   
2*sin (x)
2sin⁡2(x)2 \sin^{2}{\left(x \right)} 
                                     4             
                  /          2/x\   \              
                  |       csc |-|   |              
                  |           \2/   |     4/pi    \
                  |-1 + ------------| *csc |-- - x|
                  |        2/pi   x\|      \2     /
                  |     csc |-- - -||              
        1         \         \2    2//              
- ------------- + ---------------------------------
     /pi      \                   8/x\             
  csc|-- - 2*x|                csc |-|             
     \2       /                    \2/
(csc⁡2(x2)csc⁡2(−x2+π2)−1)4csc⁡4(−x+π2)csc⁡8(x2)−1csc⁡(−2x+π2)\frac{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right)^{4} \csc^{4}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{8}{\left(\frac{x}{2} \right)}} - \frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}} 
                    8/x\  
              16*sin |-|  
                     \2/  
-cos(2*x) + --------------
                         4
            (-1 + cos(x))
−cos⁡(2x)+16sin⁡8(x2)(cos⁡(x)−1)4- \cos{\left(2 x \right)} + \frac{16 \sin^{8}{\left(\frac{x}{2} \right)}}{\left(\cos{\left(x \right)} - 1\right)^{4}} 
     2/    pi\
2*cos |x - --|
      \    2 /
2cos⁡2(x−π2)2 \cos^{2}{\left(x - \frac{\pi}{2} \right)} 
                                           4                  
   2/    pi\      2                     cos (x)               
cos |x - --| - cos (x) + -------------------------------------
    \    2 /                2         4/    pi\      2/    pi\
                         cos (x) + cos |x - --| - cos |x - --|
                                       \    2 /       \    2 /
−cos⁡2(x)+cos⁡2(x−π2)+cos⁡4(x)cos⁡2(x)+cos⁡4(x−π2)−cos⁡2(x−π2)- \cos^{2}{\left(x \right)} + \cos^{2}{\left(x - \frac{\pi}{2} \right)} + \frac{\cos^{4}{\left(x \right)}}{\cos^{2}{\left(x \right)} + \cos^{4}{\left(x - \frac{\pi}{2} \right)} - \cos^{2}{\left(x - \frac{\pi}{2} \right)}} 
     2      
------------
   2/    pi\
sec |x - --|
    \    2 /
2sec⁡2(x−π2)\frac{2}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} 
                                           4        
                              /       2/x\\     8/x\
                              |1 - tan |-|| *cos |-|
1 - cos(2*x)   1 + cos(2*x)   \        \2//      \2/
------------ - ------------ + ----------------------
     2              2                   4           
                                     cos (x)
1−cos⁡(2x)2+(1−tan⁡2(x2))4cos⁡8(x2)cos⁡4(x)−cos⁡(2x)+12\frac{1 - \cos{\left(2 x \right)}}{2} + \frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{4} \cos^{8}{\left(\frac{x}{2} \right)}}{\cos^{4}{\left(x \right)}} - \frac{\cos{\left(2 x \right)} + 1}{2} 
                                 4             
   2         2                cos (x)          
sin (x) - cos (x) + ---------------------------
                       2         4         2   
                    cos (x) + sin (x) - sin (x)
sin⁡2(x)−cos⁡2(x)+cos⁡4(x)sin⁡4(x)−sin⁡2(x)+cos⁡2(x)\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} + \frac{\cos^{4}{\left(x \right)}}{\sin^{4}{\left(x \right)} - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}} 
                               4             
            /          2/x\   \              
            |       cos |-|   |              
            |           \2/   |     8/x   pi\
            |-1 + ------------| *cos |- - --|
            |        2/x   pi\|      \2   2 /
            |     cos |- - --||              
            \         \2   2 //              
-cos(2*x) + ---------------------------------
                            4                
                         cos (x)
(cos⁡2(x2)cos⁡2(x2−π2)−1)4cos⁡8(x2−π2)cos⁡4(x)−cos⁡(2x)\frac{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right)^{4} \cos^{8}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{4}{\left(x \right)}} - \cos{\left(2 x \right)} 
                               4        
             /         2/x\   \         
             |      sec |-|   |         
             |          \2/   |     4   
             |1 - ------------| *sec (x)
             |       2/x   pi\|         
             |    sec |- - --||         
     1       \        \2   2 //         
- -------- + ---------------------------
  sec(2*x)                8/x\          
                       sec |-|          
                           \2/
(−sec⁡2(x2)sec⁡2(x2−π2)+1)4sec⁡4(x)sec⁡8(x2)−1sec⁡(2x)\frac{\left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4} \sec^{4}{\left(x \right)}}{\sec^{8}{\left(\frac{x}{2} \right)}} - \frac{1}{\sec{\left(2 x \right)}} 
                                4        
             /        2/x   pi\\         
             |     sec |- - --||         
             |         \2   2 /|     4   
             |-1 + ------------| *sec (x)
             |          2/x\   |         
             |       sec |-|   |         
     1       \           \2/   /         
- -------- + ----------------------------
  sec(2*x)              8/x   pi\        
                     sec |- - --|        
                         \2   2 /
(−1+sec⁡2(x2−π2)sec⁡2(x2))4sec⁡4(x)sec⁡8(x2−π2)−1sec⁡(2x)\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \sec^{4}{\left(x \right)}}{\sec^{8}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(2 x \right)}} 
                         8         
            /       2/x\\     16/x\
            |1 - tan |-|| *cos  |-|
            \        \4//       \4/
-cos(2*x) + -----------------------
                       8/x\        
                    cos |-|        
                        \2/
(1−tan⁡2(x4))8cos⁡16(x4)cos⁡8(x2)−cos⁡(2x)\frac{\left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{8} \cos^{16}{\left(\frac{x}{4} \right)}}{\cos^{8}{\left(\frac{x}{2} \right)}} - \cos{\left(2 x \right)} 
                             4              8
                /       2/x\\  /       2/x\\ 
         2      |1 + tan |-|| *|1 - tan |-|| 
  1 - tan (x)   \        \2//  \        \4// 
- ----------- + -----------------------------
         2                           8       
  1 + tan (x)           /       2/x\\        
                        |1 + tan |-||        
                        \        \4//
(1−tan⁡2(x4))8(tan⁡2(x2)+1)4(tan⁡2(x4)+1)8−1−tan⁡2(x)tan⁡2(x)+1\frac{\left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{8} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} - \frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} 
-(1 - tan(x)^2)/(1 + tan(x)^2) + (1 + tan(x/2)^2)^4*(1 - tan(x/4)^2)^8/(1 + tan(x/4)^2)^8
Denominador común [src] 
                           4         8           6   
     4           2      sin (x) + sin (x) - 2*sin (x)
- sin (x) + 2*sin (x) + -----------------------------
                            2         4         2    
                         cos (x) + sin (x) - sin (x)
−sin⁡4(x)+2sin⁡2(x)+sin⁡8(x)−2sin⁡6(x)+sin⁡4(x)sin⁡4(x)−sin⁡2(x)+cos⁡2(x)- \sin^{4}{\left(x \right)} + 2 \sin^{2}{\left(x \right)} + \frac{\sin^{8}{\left(x \right)} - 2 \sin^{6}{\left(x \right)} + \sin^{4}{\left(x \right)}}{\sin^{4}{\left(x \right)} - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}} 
-sin(x)^4 + 2*sin(x)^2 + (sin(x)^4 + sin(x)^8 - 2*sin(x)^6)/(cos(x)^2 + sin(x)^4 - sin(x)^2)
    © Sr Examen – Калькулятор