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