Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
(cos(2*x))/(cos(x)-sin(x))-2*sin(x)/2*cos(x)/2
¿Cómo vas a descomponer esta (cos(2*x))/(cos(x)-sin(x))-2*sin(x)/2*cos(x)/2 expresión en fracciones?
Solución
Ha introducido
[src]
2*sin(x)
--------*cos(x)
cos(2*x) 2
--------------- - ---------------
cos(x) - sin(x) 2
$$- \frac{\frac{2 \sin{\left(x \right)}}{2} \cos{\left(x \right)}}{2} + \frac{\cos{\left(2 x \right)}}{- \sin{\left(x \right)} + \cos{\left(x \right)}}$$
cos(2*x)/(cos(x) - sin(x)) - ((2*sin(x))/2)*cos(x)/2
Simplificación general
[src]
___
cos(x)*sin(x) \/ 2 *cos(2*x)
- ------------- + --------------
2 / pi\
2*cos|x + --|
\ 4 /
$$- \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{2} + \frac{\sqrt{2} \cos{\left(2 x \right)}}{2 \cos{\left(x + \frac{\pi}{4} \right)}}$$
-cos(x)*sin(x)/2 + sqrt(2)*cos(2*x)/(2*cos(x + pi/4))
Potencias
[src]
-2*I*x 2*I*x / I*x -I*x\
e e |e e | / -I*x I*x\
------- + ------ I*|---- + -----|*\- e + e /
2 2 \ 2 2 /
--------------------------------- + ---------------------------------
I*x -I*x / -I*x I*x\ 4
e e I*\- e + e /
---- + ----- + ------------------
2 2 2
$$\frac{i \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right) \left(e^{i x} - e^{- i x}\right)}{4} + \frac{\frac{e^{2 i x}}{2} + \frac{e^{- 2 i x}}{2}}{\frac{i \left(e^{i x} - e^{- i x}\right)}{2} + \frac{e^{i x}}{2} + \frac{e^{- i x}}{2}}$$
cos(2*x) cos(x)*sin(x) ---------------- - ------------- -sin(x) + cos(x) 2
$$- \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{2} + \frac{\cos{\left(2 x \right)}}{- \sin{\left(x \right)} + \cos{\left(x \right)}}$$
cos(2*x)/(-sin(x) + cos(x)) - cos(x)*sin(x)/2
Denominador común
[src]
cos(2*x) cos(x)*sin(x) ---------------- - ------------- -sin(x) + cos(x) 2
$$- \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{2} + \frac{\cos{\left(2 x \right)}}{- \sin{\left(x \right)} + \cos{\left(x \right)}}$$
cos(2*x)/(-sin(x) + cos(x)) - cos(x)*sin(x)/2
Denominador racional
[src]
4*cos(2*x) - 2*(-sin(x) + cos(x))*cos(x)*sin(x)
-----------------------------------------------
-4*sin(x) + 4*cos(x)
$$\frac{- 2 \left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)} + 4 \cos{\left(2 x \right)}}{- 4 \sin{\left(x \right)} + 4 \cos{\left(x \right)}}$$
(4*cos(2*x) - 2*(-sin(x) + cos(x))*cos(x)*sin(x))/(-4*sin(x) + 4*cos(x))
Unión de expresiones racionales
[src]
2*cos(2*x) - (-sin(x) + cos(x))*cos(x)*sin(x)
---------------------------------------------
2*(-sin(x) + cos(x))
$$\frac{- \left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)} + 2 \cos{\left(2 x \right)}}{2 \left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right)}$$
(2*cos(2*x) - (-sin(x) + cos(x))*cos(x)*sin(x))/(2*(-sin(x) + cos(x)))
Compilar la expresión
[src]
cos(2*x) cos(x)*sin(x) ---------------- - ------------- -sin(x) + cos(x) 2
$$- \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{2} + \frac{\cos{\left(2 x \right)}}{- \sin{\left(x \right)} + \cos{\left(x \right)}}$$
cos(2*x)/(-sin(x) + cos(x)) - cos(x)*sin(x)/2
Combinatoria
[src]
2 2
2*cos(2*x) + sin (x)*cos(x) - cos (x)*sin(x)
--------------------------------------------
2*(-sin(x) + cos(x))
$$\frac{\sin^{2}{\left(x \right)} \cos{\left(x \right)} - \sin{\left(x \right)} \cos^{2}{\left(x \right)} + 2 \cos{\left(2 x \right)}}{2 \left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right)}$$
(2*cos(2*x) + sin(x)^2*cos(x) - cos(x)^2*sin(x))/(2*(-sin(x) + cos(x)))
Respuesta numérica
[src]
cos(2*x)/(-sin(x) + cos(x)) - 0.5*cos(x)*sin(x)
cos(2*x)/(-sin(x) + cos(x)) - 0.5*cos(x)*sin(x)
Parte trigonométrica
[src]
sin(2*x) ___ 2/x pi\ /x pi\
- -------- + 2*\/ 2 *sin |- + --|*cot|- + --|
4 \2 8 / \2 8 /
$$- \frac{\sin{\left(2 x \right)}}{4} + 2 \sqrt{2} \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} \cot{\left(\frac{x}{2} + \frac{\pi}{8} \right)}$$
___ /x pi\
2*\/ 2 *tan|- + --|
tan(x) \2 8 /
- --------------- + -------------------
/ 2 \ 2/x pi\
2*\1 + tan (x)/ 1 + tan |- + --|
\2 8 /
$$\frac{2 \sqrt{2} \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1} - \frac{\tan{\left(x \right)}}{2 \left(\tan^{2}{\left(x \right)} + 1\right)}$$
___ / 2/x pi\\ / 2 \
\/ 2 *|1 + cot |- + --||*\-1 + cot (x)/
cot(x) \ \2 8 //
- --------------- + ---------------------------------------
/ 2 \ / 2 \ / 2/x pi\\
2*\1 + cot (x)/ 2*\1 + cot (x)/*|-1 + cot |- + --||
\ \2 8 //
$$\frac{\sqrt{2} \left(\cot^{2}{\left(x \right)} - 1\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right)}{2 \left(\cot^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} - 1\right)} - \frac{\cot{\left(x \right)}}{2 \left(\cot^{2}{\left(x \right)} + 1\right)}$$
___ /x pi\
2*\/ 2 *tan|- + --|
sin(2*x) \2 8 /
- -------- + -------------------
4 2/x pi\
1 + tan |- + --|
\2 8 /
$$- \frac{\sin{\left(2 x \right)}}{4} + \frac{2 \sqrt{2} \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1}$$
___ / pi\
\/ 2 *cos(2*x)*sec|x + --|
sin(2*x) \ 4 /
- -------- + --------------------------
4 2
$$- \frac{\sin{\left(2 x \right)}}{4} + \frac{\sqrt{2} \cos{\left(2 x \right)} \sec{\left(x + \frac{\pi}{4} \right)}}{2}$$
___ / pi\
\/ 2 *csc|-x + --|
1 \ 4 /
- ---------- + ------------------
4*csc(2*x) /pi \
2*csc|-- - 2*x|
\2 /
$$\frac{\sqrt{2} \csc{\left(- x + \frac{\pi}{4} \right)}}{2 \csc{\left(- 2 x + \frac{\pi}{2} \right)}} - \frac{1}{4 \csc{\left(2 x \right)}}$$
sin(2*x) ___ / pi\
- -------- + \/ 2 *cos|x - --|
4 \ 4 /
$$- \frac{\sin{\left(2 x \right)}}{4} + \sqrt{2} \cos{\left(x - \frac{\pi}{4} \right)}$$
___ / 2/x pi\\ / 2 \
\/ 2 *|1 + tan |- + --||*\1 - tan (x)/
tan(x) \ \2 8 //
- --------------- + --------------------------------------
/ 2 \ / 2 \ / 2/x pi\\
2*\1 + tan (x)/ 2*\1 + tan (x)/*|1 - tan |- + --||
\ \2 8 //
$$\frac{\sqrt{2} \left(1 - \tan^{2}{\left(x \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right)}{2 \left(1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right)} - \frac{\tan{\left(x \right)}}{2 \left(\tan^{2}{\left(x \right)} + 1\right)}$$
/ pi\
cos(x)*cos|x - --|
cos(2*x) \ 2 /
---------------------- - ------------------
/ pi\ 2
- cos|x - --| + cos(x)
\ 2 /
$$- \frac{\cos{\left(x \right)} \cos{\left(x - \frac{\pi}{2} \right)}}{2} + \frac{\cos{\left(2 x \right)}}{\cos{\left(x \right)} - \cos{\left(x - \frac{\pi}{2} \right)}}$$
/ 2/x\\ /x\
2 |-1 + cot |-||*cot|-|
-1 + cot (x) \ \2// \2/
------------------------------------------ - ---------------------
/ 2/x\ /x\ \ 2
|-1 + cot |-| 2*cot|-| | / 2/x\\
/ 2 \ | \2/ \2/ | |1 + cot |-||
\1 + cot (x)/*|------------ - -----------| \ \2//
| 2/x\ 2/x\|
|1 + cot |-| 1 + cot |-||
\ \2/ \2//
$$- \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \cot{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{\cot^{2}{\left(x \right)} - 1}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(x \right)} + 1\right)}$$
/pi \ / pi\
sin|-- + 2*x| sin(x)*sin|x + --|
\2 / \ 2 /
--------------------- - ------------------
/ pi\ 2
-sin(x) + sin|x + --|
\ 2 /
$$- \frac{\sin{\left(x \right)} \sin{\left(x + \frac{\pi}{2} \right)}}{2} + \frac{\sin{\left(2 x + \frac{\pi}{2} \right)}}{- \sin{\left(x \right)} + \sin{\left(x + \frac{\pi}{2} \right)}}$$
___ / pi\
\/ 2 *cos(2*x)*csc|-x + --|
sin(2*x) \ 4 /
- -------- + ---------------------------
4 2
$$- \frac{\sin{\left(2 x \right)}}{4} + \frac{\sqrt{2} \cos{\left(2 x \right)} \csc{\left(- x + \frac{\pi}{4} \right)}}{2}$$
___
1 \/ 2
- --------------- + -----------
/ pi\ / pi\
4*sec|2*x - --| sec|x - --|
\ 2 / \ 4 /
$$- \frac{1}{4 \sec{\left(2 x - \frac{\pi}{2} \right)}} + \frac{\sqrt{2}}{\sec{\left(x - \frac{\pi}{4} \right)}}$$
___ / pi\
\/ 2 *sec|x + --|
1 \ 4 /
- --------------- + -----------------
/ pi\ 2*sec(2*x)
4*sec|2*x - --|
\ 2 /
$$- \frac{1}{4 \sec{\left(2 x - \frac{\pi}{2} \right)}} + \frac{\sqrt{2} \sec{\left(x + \frac{\pi}{4} \right)}}{2 \sec{\left(2 x \right)}}$$
1 1 -------------------------- - --------------- / 1 1 \ 2*csc(x)*sec(x) |------ - ------|*sec(2*x) \sec(x) csc(x)/
$$- \frac{1}{2 \csc{\left(x \right)} \sec{\left(x \right)}} + \frac{1}{\left(\frac{1}{\sec{\left(x \right)}} - \frac{1}{\csc{\left(x \right)}}\right) \sec{\left(2 x \right)}}$$
___
sin(2*x) \/ 2 *cos(2*x)
- -------- + --------------
4 / pi\
2*cos|x + --|
\ 4 /
$$- \frac{\sin{\left(2 x \right)}}{4} + \frac{\sqrt{2} \cos{\left(2 x \right)}}{2 \cos{\left(x + \frac{\pi}{4} \right)}}$$
1 1 ------------------------------- - -------------------- / 1 1 \ / pi\ |------ - -----------|*sec(2*x) 2*sec(x)*sec|x - --| |sec(x) / pi\| \ 2 / | sec|x - --|| \ \ 2 //
$$- \frac{1}{2 \sec{\left(x \right)} \sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\left(- \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(2 x \right)}}$$
sin(2*x) ___ / pi\
- -------- + \/ 2 *sin|x + --|
4 \ 4 /
$$- \frac{\sin{\left(2 x \right)}}{4} + \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}$$
___ /pi \
\/ 2 *sin|-- + 2*x|
sin(2*x) \2 /
- -------- + -------------------
4 / 3*pi\
2*sin|x + ----|
\ 4 /
$$- \frac{\sin{\left(2 x \right)}}{4} + \frac{\sqrt{2} \sin{\left(2 x + \frac{\pi}{2} \right)}}{2 \sin{\left(x + \frac{3 \pi}{4} \right)}}$$
___ 2 / 2/x pi\\ / 2 \
\/ 2 *cos (x)*|1 + tan |- + --||*\1 - tan (x)/
sin(2*x) \ \2 8 //
- -------- + ----------------------------------------------
4 / 2/x pi\\
2*|1 - tan |- + --||
\ \2 8 //
$$\frac{\sqrt{2} \left(1 - \tan^{2}{\left(x \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right) \cos^{2}{\left(x \right)}}{2 \left(1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}\right)} - \frac{\sin{\left(2 x \right)}}{4}$$
___
1 \/ 2
- ---------- + -----------
4*csc(2*x) / pi\
csc|x + --|
\ 4 /
$$\frac{\sqrt{2}}{\csc{\left(x + \frac{\pi}{4} \right)}} - \frac{1}{4 \csc{\left(2 x \right)}}$$
___ /x pi\
2*\/ 2 *cot|- + --|
cot(x) \2 8 /
- --------------- + -------------------
/ 2 \ 2/x pi\
2*\1 + cot (x)/ 1 + cot |- + --|
\2 8 /
$$\frac{2 \sqrt{2} \cot{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1} - \frac{\cot{\left(x \right)}}{2 \left(\cot^{2}{\left(x \right)} + 1\right)}$$
___
sin(2*x) \/ 2 *cos(2*x)
- -------- - -----------------------------------------------------
4 ___________ ___________
___ / ___ / ___
- \/ 2 *cos(x) + \/ 2 + \/ 2 *\/ 2 - \/ 2 *sin(x)
$$- \frac{\sin{\left(2 x \right)}}{4} - \frac{\sqrt{2} \cos{\left(2 x \right)}}{\sqrt{2 - \sqrt{2}} \sqrt{\sqrt{2} + 2} \sin{\left(x \right)} - \sqrt{2} \cos{\left(x \right)}}$$
/ 2/x\\ /x\
2 |1 - tan |-||*tan|-|
1 - tan (x) \ \2// \2/
----------------------------------------- - --------------------
/ 2/x\ /x\ \ 2
|1 - tan |-| 2*tan|-| | / 2/x\\
/ 2 \ | \2/ \2/ | |1 + tan |-||
\1 + tan (x)/*|----------- - -----------| \ \2//
| 2/x\ 2/x\|
|1 + tan |-| 1 + tan |-||
\ \2/ \2//
$$- \frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \tan{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{1 - \tan^{2}{\left(x \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}$$
/ pi\
cos|2*x - --|
\ 2 / ___ / pi\
- ------------- + \/ 2 *cos|x - --|
4 \ 4 /
$$\sqrt{2} \cos{\left(x - \frac{\pi}{4} \right)} - \frac{\cos{\left(2 x - \frac{\pi}{2} \right)}}{4}$$
/ pi\
cos|2*x - --| ___
\ 2 / \/ 2 *cos(2*x)
- ------------- + --------------
4 / pi\
2*cos|x + --|
\ 4 /
$$\frac{\sqrt{2} \cos{\left(2 x \right)}}{2 \cos{\left(x + \frac{\pi}{4} \right)}} - \frac{\cos{\left(2 x - \frac{\pi}{2} \right)}}{4}$$
1 1 ------------------------------------ - -------------------- / 1 1 \ /pi \ /pi \ |----------- - ------|*csc|-- - 2*x| 2*csc(x)*csc|-- - x| | /pi \ csc(x)| \2 / \2 / |csc|-- - x| | \ \2 / /
$$- \frac{1}{2 \csc{\left(x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(- 2 x + \frac{\pi}{2} \right)}}$$
cos(2*x) cos(x)*sin(x) ---------------- - ------------- -sin(x) + cos(x) 2
$$- \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{2} + \frac{\cos{\left(2 x \right)}}{- \sin{\left(x \right)} + \cos{\left(x \right)}}$$
cos(2*x)/(-sin(x) + cos(x)) - cos(x)*sin(x)/2