Sr Examen
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?
Solución
Ha introducido
[src]
4
2 2 cos (x)
sin (x) - cos (x) + ---------------------------
2 2 4
cos (x) - sin (x) + sin (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)
Potencias
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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
$$\frac{\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{4}}{\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2} + \frac{\left(e^{i x} - e^{- i x}\right)^{4}}{16} + \frac{\left(e^{i x} - e^{- i x}\right)^{2}}{4}} - \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2} - \frac{\left(e^{i x} - e^{- i x}\right)^{2}}{4}$$
4
2 2 cos (x)
sin (x) - cos (x) + ---------------------------
2 4 2
cos (x) + sin (x) - sin (x)
$$\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} + \frac{\cos^{4}{\left(x \right)}}{\sin^{4}{\left(x \right)} - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}$$
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)
$$\frac{\left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \left(\sin^{4}{\left(x \right)} - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) + \cos^{4}{\left(x \right)}}{\sin^{4}{\left(x \right)} - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}$$
(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
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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)
$$- \frac{\left(- \sin^{4}{\left(x \right)} + \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} + \sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)}}{\sin^{4}{\left(x \right)} - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}$$
-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)
$$\frac{\left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \left(\sin^{4}{\left(x \right)} - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) + \cos^{4}{\left(x \right)}}{\sin^{4}{\left(x \right)} - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}$$
(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)
$$\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} + \frac{\cos^{4}{\left(x \right)}}{\sin^{4}{\left(x \right)} - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}$$
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// /
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{4}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4} \left(\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} - \frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{16 \tan^{4}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}}\right)} - \frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
1 1 1
------- - ------- + -------------------------------------
2 2 / 1 1 1 \ 4
csc (x) sec (x) |------- + ------- - -------|*sec (x)
| 4 2 2 |
\csc (x) sec (x) csc (x)/
$$- \frac{1}{\sec^{2}{\left(x \right)}} + \frac{1}{\csc^{2}{\left(x \right)}} + \frac{1}{\left(\frac{1}{\sec^{2}{\left(x \right)}} - \frac{1}{\csc^{2}{\left(x \right)}} + \frac{1}{\csc^{4}{\left(x \right)}}\right) \sec^{4}{\left(x \right)}}$$
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//
$$\frac{256 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4} \tan^{8}{\left(\frac{x}{4} \right)}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{4} \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} - \frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}$$
4
/ 2/x pi\\
| cos |- - --||
| \2 2 /| 8/x\
|1 - ------------| *cos |-|
| 2/x\ | \2/
| cos |-| |
\ \2/ /
-cos(2*x) + ---------------------------
4
cos (x)
$$\frac{\left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \cos^{8}{\left(\frac{x}{2} \right)}}{\cos^{4}{\left(x \right)}} - \cos{\left(2 x \right)}$$
4
/ 2/x\\ 8/x\ 4
-cos(2*x) + |1 - tan |-|| *cos |-|*sec (x)
\ \2// \2/
$$\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{4} \cos^{8}{\left(\frac{x}{2} \right)} \sec^{4}{\left(x \right)} - \cos{\left(2 x \right)}$$
4
/ 2 \
| sin (x) | 8/x\
|-1 + ---------| *sin |-|
| 4/x\| \2/
| 4*sin |-||
\ \2//
-cos(2*x) + -------------------------
4
cos (x)
$$\frac{\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)}}{\cos^{4}{\left(x \right)}} - \cos{\left(2 x \right)}$$
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//
$$- \frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} + \frac{256 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4} \cot^{8}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}}$$
1 - cos(2*x)
$$1 - \cos{\left(2 x \right)}$$
2 ------- 2 csc (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 /
$$\sin^{2}{\left(x \right)} - \sin^{2}{\left(x + \frac{\pi}{2} \right)} + \frac{\sin^{4}{\left(x + \frac{\pi}{2} \right)}}{\sin^{4}{\left(x \right)} - \sin^{2}{\left(x \right)} + \sin^{2}{\left(x + \frac{\pi}{2} \right)}}$$
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/
$$\frac{\left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \csc^{4}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{8}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}$$
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 /
$$\frac{\left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{4} \sin^{8}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{\sin^{4}{\left(x + \frac{\pi}{2} \right)}} - \sin{\left(2 x + \frac{\pi}{2} \right)}$$
2/x\
8*cot |-|
\2/
--------------
2
/ 2/x\\
|1 + cot |-||
\ \2//
$$\frac{8 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
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// /
$$\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4} \left(\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} - \frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{16 \cot^{4}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}}\right)} - \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
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//
$$\frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{8} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4}} - \frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1}$$
8 8
/ /x\\ / /x\\ 8/x\
|1 + tan|-|| *|-1 + tan|-|| *tan |-|
\ \4// \ \4// \2/
-cos(2*x) + ------------------------------------
8/x\
256*tan |-|
\4/
$$\frac{\left(\tan{\left(\frac{x}{4} \right)} - 1\right)^{8} \left(\tan{\left(\frac{x}{4} \right)} + 1\right)^{8} \tan^{8}{\left(\frac{x}{2} \right)}}{256 \tan^{8}{\left(\frac{x}{4} \right)}} - \cos{\left(2 x \right)}$$
2/x\
8*tan |-|
\2/
--------------
2
/ 2/x\\
|1 + tan |-||
\ \2//
$$\frac{8 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
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 / /
$$- \frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{2}{\left(x \right)}} + \frac{1}{\left(\frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} - \frac{1}{\csc^{2}{\left(x \right)}} + \frac{1}{\csc^{4}{\left(x \right)}}\right) \csc^{4}{\left(- x + \frac{\pi}{2} \right)}}$$
4
/ 1 \ 8/x\
|-1 + -------| *sin |-|
| 2/x\| \2/
| tan |-||
\ \2//
-cos(2*x) + -----------------------
4
cos (x)
$$\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)}}{\cos^{4}{\left(x \right)}} - \cos{\left(2 x \right)}$$
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 //
$$\frac{1}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} - \frac{1}{\sec^{2}{\left(x \right)}} + \frac{1}{\left(- \frac{1}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec^{4}{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec^{2}{\left(x \right)}}\right) \sec^{4}{\left(x \right)}}$$
4
/ 2/x\\ 8/x\
|-1 + cot |-|| *sin |-|
1 - cos(2*x) 1 + cos(2*x) \ \2// \2/
------------ - ------------ + -----------------------
2 2 4
cos (x)
$$\frac{1 - \cos{\left(2 x \right)}}{2} - \frac{\cos{\left(2 x \right)} + 1}{2} + \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)}}{\cos^{4}{\left(x \right)}}$$
4
/ 2 \
| sin (x) | 8/x\
|-1 + ---------| *sin |-|
| 4/x\| \2/
| 4*sin |-||
/pi \ \ \2//
- sin|-- + 2*x| + -------------------------
\2 / 4/ pi\
sin |x + --|
\ 2 /
$$\frac{\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)}}{\sin^{4}{\left(x + \frac{\pi}{2} \right)}} - \sin{\left(2 x + \frac{\pi}{2} \right)}$$
2 2*sin (x)
$$2 \sin^{2}{\left(x \right)}$$
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/
$$\frac{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right)^{4} \csc^{4}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{8}{\left(\frac{x}{2} \right)}} - \frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}$$
8/x\
16*sin |-|
\2/
-cos(2*x) + --------------
4
(-1 + cos(x))
$$- \cos{\left(2 x \right)} + \frac{16 \sin^{8}{\left(\frac{x}{2} \right)}}{\left(\cos{\left(x \right)} - 1\right)^{4}}$$
2/ pi\
2*cos |x - --|
\ 2 /
$$2 \cos^{2}{\left(x - \frac{\pi}{2} \right)}$$
4
2/ pi\ 2 cos (x)
cos |x - --| - cos (x) + -------------------------------------
\ 2 / 2 4/ pi\ 2/ pi\
cos (x) + cos |x - --| - cos |x - --|
\ 2 / \ 2 /
$$- \cos^{2}{\left(x \right)} + \cos^{2}{\left(x - \frac{\pi}{2} \right)} + \frac{\cos^{4}{\left(x \right)}}{\cos^{2}{\left(x \right)} + \cos^{4}{\left(x - \frac{\pi}{2} \right)} - \cos^{2}{\left(x - \frac{\pi}{2} \right)}}$$
2
------------
2/ pi\
sec |x - --|
\ 2 /
$$\frac{2}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}$$
4
/ 2/x\\ 8/x\
|1 - tan |-|| *cos |-|
1 - cos(2*x) 1 + cos(2*x) \ \2// \2/
------------ - ------------ + ----------------------
2 2 4
cos (x)
$$\frac{1 - \cos{\left(2 x \right)}}{2} + \frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{4} \cos^{8}{\left(\frac{x}{2} \right)}}{\cos^{4}{\left(x \right)}} - \frac{\cos{\left(2 x \right)} + 1}{2}$$
4
2 2 cos (x)
sin (x) - cos (x) + ---------------------------
2 4 2
cos (x) + sin (x) - sin (x)
$$\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} + \frac{\cos^{4}{\left(x \right)}}{\sin^{4}{\left(x \right)} - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}$$
4
/ 2/x\ \
| cos |-| |
| \2/ | 8/x pi\
|-1 + ------------| *cos |- - --|
| 2/x pi\| \2 2 /
| cos |- - --||
\ \2 2 //
-cos(2*x) + ---------------------------------
4
cos (x)
$$\frac{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right)^{4} \cos^{8}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{4}{\left(x \right)}} - \cos{\left(2 x \right)}$$
4
/ 2/x\ \
| sec |-| |
| \2/ | 4
|1 - ------------| *sec (x)
| 2/x pi\|
| sec |- - --||
1 \ \2 2 //
- -------- + ---------------------------
sec(2*x) 8/x\
sec |-|
\2/
$$\frac{\left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4} \sec^{4}{\left(x \right)}}{\sec^{8}{\left(\frac{x}{2} \right)}} - \frac{1}{\sec{\left(2 x \right)}}$$
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 /
$$\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \sec^{4}{\left(x \right)}}{\sec^{8}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(2 x \right)}}$$
8
/ 2/x\\ 16/x\
|1 - tan |-|| *cos |-|
\ \4// \4/
-cos(2*x) + -----------------------
8/x\
cos |-|
\2/
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{8} \cos^{16}{\left(\frac{x}{4} \right)}}{\cos^{8}{\left(\frac{x}{2} \right)}} - \cos{\left(2 x \right)}$$
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//
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{8} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} - \frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}$$
-(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)
$$- \sin^{4}{\left(x \right)} + 2 \sin^{2}{\left(x \right)} + \frac{\sin^{8}{\left(x \right)} - 2 \sin^{6}{\left(x \right)} + \sin^{4}{\left(x \right)}}{\sin^{4}{\left(x \right)} - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}$$
-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)