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