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