Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
cos(x)/(1+sin(x))-sin(x)/(1-sin(x))
¿Cómo vas a descomponer esta cos(x)/(1+sin(x))-sin(x)/(1-sin(x)) expresión en fracciones?
Solución
Ha introducido
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cos(x) sin(x) ---------- - ---------- 1 + sin(x) 1 - sin(x)
$$\frac{\cos{\left(x \right)}}{\sin{\left(x \right)} + 1} - \frac{\sin{\left(x \right)}}{1 - \sin{\left(x \right)}}$$
cos(x)/(1 + sin(x)) - sin(x)/(1 - sin(x))
Simplificación general
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cos(x) sin(x) ---------- + ----------- 1 + sin(x) -1 + sin(x)
$$\frac{\cos{\left(x \right)}}{\sin{\left(x \right)} + 1} + \frac{\sin{\left(x \right)}}{\sin{\left(x \right)} - 1}$$
cos(x)/(1 + sin(x)) + sin(x)/(-1 + sin(x))
Respuesta numérica
[src]
cos(x)/(1.0 + sin(x)) - sin(x)/(1.0 - sin(x))
cos(x)/(1.0 + sin(x)) - sin(x)/(1.0 - sin(x))
Potencias
[src]
I*x -I*x
e e
---- + ----- / -I*x I*x\
2 2 I*\- e + e /
---------------------- + --------------------------
/ -I*x I*x\ / / -I*x I*x\\
I*\- e + e / | I*\- e + e /|
1 - ------------------ 2*|1 + ------------------|
2 \ 2 /
$$\frac{i \left(e^{i x} - e^{- i x}\right)}{2 \left(\frac{i \left(e^{i x} - e^{- i x}\right)}{2} + 1\right)} + \frac{\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}}{- \frac{i \left(e^{i x} - e^{- i x}\right)}{2} + 1}$$
(exp(i*x)/2 + exp(-i*x)/2)/(1 - i*(-exp(-i*x) + exp(i*x))/2) + i*(-exp(-i*x) + exp(i*x))/(2*(1 + i*(-exp(-i*x) + exp(i*x))/2))
Combinatoria
[src]
2
sin (x) - cos(x) + cos(x)*sin(x) + sin(x)
-----------------------------------------
(1 + sin(x))*(-1 + sin(x))
$$\frac{\sin^{2}{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} - \cos{\left(x \right)}}{\left(\sin{\left(x \right)} - 1\right) \left(\sin{\left(x \right)} + 1\right)}$$
(sin(x)^2 - cos(x) + cos(x)*sin(x) + sin(x))/((1 + sin(x))*(-1 + sin(x)))
Unión de expresiones racionales
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(1 - sin(x))*cos(x) - (1 + sin(x))*sin(x)
-----------------------------------------
(1 - sin(x))*(1 + sin(x))
$$\frac{\left(1 - \sin{\left(x \right)}\right) \cos{\left(x \right)} - \left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\left(1 - \sin{\left(x \right)}\right) \left(\sin{\left(x \right)} + 1\right)}$$
((1 - sin(x))*cos(x) - (1 + sin(x))*sin(x))/((1 - sin(x))*(1 + sin(x)))
Denominador común
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1 - cos(x) + cos(x)*sin(x) + sin(x)
1 + -----------------------------------
2
-1 + sin (x)
$$1 + \frac{\sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} - \cos{\left(x \right)} + 1}{\sin^{2}{\left(x \right)} - 1}$$
1 + (1 - cos(x) + cos(x)*sin(x) + sin(x))/(-1 + sin(x)^2)
Denominador racional
[src]
(1 - sin(x))*cos(x) - (1 + sin(x))*sin(x)
-----------------------------------------
(1 - sin(x))*(1 + sin(x))
$$\frac{\left(1 - \sin{\left(x \right)}\right) \cos{\left(x \right)} - \left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\left(1 - \sin{\left(x \right)}\right) \left(\sin{\left(x \right)} + 1\right)}$$
((1 - sin(x))*cos(x) - (1 + sin(x))*sin(x))/((1 - sin(x))*(1 + sin(x)))
Parte trigonométrica
[src]
2/x\ /x\
-1 + cot |-| 2*cot|-|
\2/ \2/
------------------------------- - -------------------------------
/ /x\ \ / /x\ \
| 2*cot|-| | | 2*cot|-| |
/ 2/x\\ | \2/ | / 2/x\\ | \2/ |
|1 + cot |-||*|1 + -----------| |1 + cot |-||*|1 - -----------|
\ \2// | 2/x\| \ \2// | 2/x\|
| 1 + cot |-|| | 1 + cot |-||
\ \2// \ \2//
$$\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} - \frac{2 \cot{\left(\frac{x}{2} \right)}}{\left(1 - \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
___ 2/ pi\ / 2/ pi\\ ___ 2/x pi\ / 2/x pi\\
-1 + \/ 2 *sin |x + --|*|-1 + cot |x + --|| + 2*\/ 2 *sin |- + --|*|-1 + cot |- + --||
\ 8 / \ \ 8 // \2 8 / \ \2 8 //
--------------------------------------------------------------------------------------
2
2*cos (x)
$$\frac{2 \sqrt{2} \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} - 1\right) \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + \sqrt{2} \left(\cot^{2}{\left(x + \frac{\pi}{8} \right)} - 1\right) \sin^{2}{\left(x + \frac{\pi}{8} \right)} - 1}{2 \cos^{2}{\left(x \right)}}$$
2/x\
1 - 2*sin |-|
\2/ sin(x)
------------- + -----------
1 + sin(x) -1 + sin(x)
$$\frac{1 - 2 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)} + 1} + \frac{\sin{\left(x \right)}}{\sin{\left(x \right)} - 1}$$
/ pi\ sin|x + --| \ 2 / sin(x) ----------- - ---------- 1 + sin(x) 1 - sin(x)
$$\frac{\sin{\left(x + \frac{\pi}{2} \right)}}{\sin{\left(x \right)} + 1} - \frac{\sin{\left(x \right)}}{1 - \sin{\left(x \right)}}$$
2/x pi\ / pi\
1 - 2*cos |- - --| cos|x - --|
\2 2 / \ 2 /
------------------ + ----------------
/ pi\ / pi\
1 + cos|x - --| -1 + cos|x - --|
\ 2 / \ 2 /
$$\frac{1 - 2 \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(x - \frac{\pi}{2} \right)} + 1} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x - \frac{\pi}{2} \right)} - 1}$$
1 1 ------------------- - ------------------- / 1 \ / 1 \ |1 + ------|*sec(x) |1 - ------|*csc(x) \ csc(x)/ \ csc(x)/
$$\frac{1}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right) \sec{\left(x \right)}} - \frac{1}{\left(1 - \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)}}$$
1 1 ------------------------ - ----------------------------- / 1 \ / 1 \ / pi\ |1 + -----------|*sec(x) |1 - -----------|*sec|x - --| | / pi\| | / pi\| \ 2 / | sec|x - --|| | sec|x - --|| \ \ 2 // \ \ 2 //
$$\frac{1}{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \sec{\left(x \right)}} - \frac{1}{\left(1 - \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}}$$
2/x\ /x\
1 - tan |-| 2*tan|-|
\2/ \2/
------------------------------- - -------------------------------
/ /x\ \ / /x\ \
| 2*tan|-| | | 2*tan|-| |
/ 2/x\\ | \2/ | / 2/x\\ | \2/ |
|1 + tan |-||*|1 + -----------| |1 + tan |-||*|1 - -----------|
\ \2// | 2/x\| \ \2// | 2/x\|
| 1 + tan |-|| | 1 + tan |-||
\ \2// \ \2//
$$\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} - \frac{2 \tan{\left(\frac{x}{2} \right)}}{\left(1 - \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
___ / 2/ pi\\ ___ / 2/x pi\\
\/ 2 *|1 - tan |x + --|| 2*\/ 2 *|1 - tan |- + --||
\ \ 8 // \ \2 8 //
-1 + ------------------------ + --------------------------
2/ pi\ 2/x pi\
1 + tan |x + --| 1 + tan |- + --|
\ 8 / \2 8 /
----------------------------------------------------------
2
1 - tan (x)
1 + -----------
2
1 + tan (x)
$$\frac{\frac{2 \sqrt{2} \left(1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1} + \frac{\sqrt{2} \left(1 - \tan^{2}{\left(x + \frac{\pi}{8} \right)}\right)}{\tan^{2}{\left(x + \frac{\pi}{8} \right)} + 1} - 1}{\frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} + 1}$$
2
1 - -------
2/x\
csc |-|
\2/ 1
----------- + --------------------
1 / 1 \
1 + ------ |-1 + ------|*csc(x)
csc(x) \ csc(x)/
$$\frac{1 - \frac{2}{\csc^{2}{\left(\frac{x}{2} \right)}}}{1 + \frac{1}{\csc{\left(x \right)}}} + \frac{1}{\left(-1 + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)}}$$
1 1 ------------------------ - ------------------- / 1 \ /pi \ / 1 \ |1 + ------|*csc|-- - x| |1 - ------|*csc(x) \ csc(x)/ \2 / \ csc(x)/
$$\frac{1}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(- x + \frac{\pi}{2} \right)}} - \frac{1}{\left(1 - \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)}}$$
2/x\
8*tan |-|
\4/
1 - --------------
2
/ 2/x\\ /x\
|1 + tan |-|| 2*tan|-|
\ \4// \2/
------------------ + --------------------------------
/x\ / /x\ \
2*tan|-| | 2*tan|-| |
\2/ / 2/x\\ | \2/ |
1 + ----------- |1 + tan |-||*|-1 + -----------|
2/x\ \ \2// | 2/x\|
1 + tan |-| | 1 + tan |-||
\2/ \ \2//
$$\frac{1 - \frac{8 \tan^{2}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}}{1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\left(-1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
___ / pi\ ___ / pi\
-1 + \/ 2 *cos|2*x + --| + 2*\/ 2 *cos|x + --|
\ 4 / \ 4 /
----------------------------------------------
1 + cos(2*x)
$$\frac{2 \sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)} + \sqrt{2} \cos{\left(2 x + \frac{\pi}{4} \right)} - 1}{\cos{\left(2 x \right)} + 1}$$
/ pi\
cos|x - --|
cos(x) \ 2 /
--------------- - ---------------
/ pi\ / pi\
1 + cos|x - --| 1 - cos|x - --|
\ 2 / \ 2 /
$$\frac{\cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)} + 1} - \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{1 - \cos{\left(x - \frac{\pi}{2} \right)}}$$
___ / 2/ pi\\ ___ / 2/x pi\\
\/ 2 *|-1 + cot |x + --|| 2*\/ 2 *|-1 + cot |- + --||
\ \ 8 // \ \2 8 //
-1 + ------------------------- + ---------------------------
2/ pi\ 2/x pi\
1 + cot |x + --| 1 + cot |- + --|
\ 8 / \2 8 /
------------------------------------------------------------
2
-1 + cot (x)
1 + ------------
2
1 + cot (x)
$$\frac{\frac{2 \sqrt{2} \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1} + \frac{\sqrt{2} \left(\cot^{2}{\left(x + \frac{\pi}{8} \right)} - 1\right)}{\cot^{2}{\left(x + \frac{\pi}{8} \right)} + 1} - 1}{\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} + 1}$$
___ ___
\/ 2 2*\/ 2
-1 + -------------- + ------------
/ pi\ / pi\
csc|-2*x + --| csc|-x + --|
\ 4 / \ 4 /
----------------------------------
1
1 + -------------
/pi \
csc|-- - 2*x|
\2 /
$$\frac{-1 + \frac{2 \sqrt{2}}{\csc{\left(- x + \frac{\pi}{4} \right)}} + \frac{\sqrt{2}}{\csc{\left(- 2 x + \frac{\pi}{4} \right)}}}{1 + \frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}}$$
2
1 - ------------
2/x pi\
sec |- - --|
\2 2 / 1
---------------- + ------------------------------
1 / 1 \ / pi\
1 + ----------- |-1 + -----------|*sec|x - --|
/ pi\ | / pi\| \ 2 /
sec|x - --| | sec|x - --||
\ 2 / \ \ 2 //
$$\frac{1 - \frac{2}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}}{1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}} + \frac{1}{\left(-1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}}$$
___ ___
\/ 2 2*\/ 2
-1 + ------------- + -----------
/ pi\ / pi\
sec|2*x + --| sec|x + --|
\ 4 / \ 4 /
--------------------------------
1
1 + --------
sec(2*x)
$$\frac{-1 + \frac{\sqrt{2}}{\sec{\left(2 x + \frac{\pi}{4} \right)}} + \frac{2 \sqrt{2}}{\sec{\left(x + \frac{\pi}{4} \right)}}}{1 + \frac{1}{\sec{\left(2 x \right)}}}$$
2/x\
8*cot |-|
\4/
1 - --------------
2
/ 2/x\\ /x\
|1 + cot |-|| 2*cot|-|
\ \4// \2/
------------------ + --------------------------------
/x\ / /x\ \
2*cot|-| | 2*cot|-| |
\2/ / 2/x\\ | \2/ |
1 + ----------- |1 + cot |-||*|-1 + -----------|
2/x\ \ \2// | 2/x\|
1 + cot |-| | 1 + cot |-||
\2/ \ \2//
$$\frac{1 - \frac{8 \cot^{2}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}}{1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\left(-1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
___ / 2/ pi\\ ___ / 2/x pi\\
\/ 2 *|1 - tan |x + --|| 2*\/ 2 *|1 - tan |- + --||
\ \ 8 // \ \2 8 //
-1 + ------------------------ + --------------------------
2/ pi\ 2/x pi\
1 + tan |x + --| 1 + tan |- + --|
\ 8 / \2 8 /
----------------------------------------------------------
2
2*cos (x)
$$\frac{\frac{2 \sqrt{2} \left(1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1} + \frac{\sqrt{2} \left(1 - \tan^{2}{\left(x + \frac{\pi}{8} \right)}\right)}{\tan^{2}{\left(x + \frac{\pi}{8} \right)} + 1} - 1}{2 \cos^{2}{\left(x \right)}}$$
___ / 3*pi\ ___ / 3*pi\
-1 + \/ 2 *sin|2*x + ----| + 2*\/ 2 *sin|x + ----|
\ 4 / \ 4 /
--------------------------------------------------
/pi \
1 + sin|-- + 2*x|
\2 /
$$\frac{2 \sqrt{2} \sin{\left(x + \frac{3 \pi}{4} \right)} + \sqrt{2} \sin{\left(2 x + \frac{3 \pi}{4} \right)} - 1}{\sin{\left(2 x + \frac{\pi}{2} \right)} + 1}$$
___ / pi\ ___ / pi\
-1 + \/ 2 *cos|2*x + --| + 2*\/ 2 *cos|x + --|
\ 4 / \ 4 /
----------------------------------------------
2
2*cos (x)
$$\frac{2 \sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)} + \sqrt{2} \cos{\left(2 x + \frac{\pi}{4} \right)} - 1}{2 \cos^{2}{\left(x \right)}}$$
(-1 + sqrt(2)*cos(2*x + pi/4) + 2*sqrt(2)*cos(x + pi/4))/(2*cos(x)^2)