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