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