Sr Examen
¿Cómo vas a descomponer esta tg(a)cos(a)/(1+ctg^2(a)) expresión en fracciones?
Solución
Ha introducido
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tan(a)*cos(a)
-------------
2
1 + cot (a)
$$\frac{\cos{\left(a \right)} \tan{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1}$$
(tan(a)*cos(a))/(1 + cot(a)^2)
Potencias
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/ I*a -I*a\ |e e | / I*a -I*a\ I*|---- + -----|*\- e + e / \ 2 2 / --------------------------------- / 2 \ / I*a -I*a\ \1 + cot (a)/*\e + e /
$$\frac{i \left(- e^{i a} + e^{- i a}\right) \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)}{\left(e^{i a} + e^{- i a}\right) \left(\cot^{2}{\left(a \right)} + 1\right)}$$
i*(exp(i*a)/2 + exp(-i*a)/2)*(-exp(i*a) + exp(-i*a))/((1 + cot(a)^2)*(exp(i*a) + exp(-i*a)))
Parte trigonométrica
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2
2*sin (a)*cos(a)
------------------------
/ 2 \
| sin (2*a)|
|1 + ---------|*sin(2*a)
| 4 |
\ 4*sin (a)/
$$\frac{2 \sin^{2}{\left(a \right)} \cos{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}\right) \sin{\left(2 a \right)}}$$
-sin(4*a) + 2*sin(2*a)
----------------------
/ pi\
8*sin|a + --|
\ 2 /
$$\frac{2 \sin{\left(2 a \right)} - \sin{\left(4 a \right)}}{8 \sin{\left(a + \frac{\pi}{2} \right)}}$$
/ pi\
cos|a - --|
\ 2 /
----------------
2
cos (a)
1 + ------------
2/ pi\
cos |a - --|
\ 2 /
$$\frac{\cos{\left(a - \frac{\pi}{2} \right)}}{\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} + 1}$$
sin(3*a) 3*sin(a)
- -------- + --------
4 4
$$\frac{3 \sin{\left(a \right)}}{4} - \frac{\sin{\left(3 a \right)}}{4}$$
/ pi\ / pi\
- cos|4*a - --| + 2*cos|2*a - --|
\ 2 / \ 2 /
---------------------------------
8*cos(a)
$$\frac{2 \cos{\left(2 a - \frac{\pi}{2} \right)} - \cos{\left(4 a - \frac{\pi}{2} \right)}}{8 \cos{\left(a \right)}}$$
-sin(4*a) + 2*sin(2*a)
----------------------
8*cos(a)
$$\frac{2 \sin{\left(2 a \right)} - \sin{\left(4 a \right)}}{8 \cos{\left(a \right)}}$$
1 ------- 3 csc (a)
$$\frac{1}{\csc^{3}{\left(a \right)}}$$
1 ------------------------------ / 2/ pi\\ | sec |a - --|| | \ 2 /| / pi\ |1 + ------------|*sec|a - --| | 2 | \ 2 / \ sec (a) /
$$\frac{1}{\left(1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}\right) \sec{\left(a - \frac{\pi}{2} \right)}}$$
sin(a)
-----------
2
cos (a)
1 + -------
2
sin (a)
$$\frac{\sin{\left(a \right)}}{1 + \frac{\cos^{2}{\left(a \right)}}{\sin^{2}{\left(a \right)}}}$$
/ 2/a\\
|1 - tan |-||*tan(a)
\ \2//
---------------------------
/ 1 \ / 2/a\\
|1 + -------|*|1 + tan |-||
| 2 | \ \2//
\ tan (a)/
$$\frac{\left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right) \tan{\left(a \right)}}{\left(1 + \frac{1}{\tan^{2}{\left(a \right)}}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
/ 2/a\\ / 2*tan(2*a) 4*tan(a) \
|1 + tan |-||*|- ------------- + -----------|
\ \2// | 2 2 |
\ 1 + tan (2*a) 1 + tan (a)/
---------------------------------------------
/ 2/a\\
8*|1 - tan |-||
\ \2//
$$\frac{\left(- \frac{2 \tan{\left(2 a \right)}}{\tan^{2}{\left(2 a \right)} + 1} + \frac{4 \tan{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}{8 \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)}$$
3/a\
8*cot |-|
\2/
--------------
3
/ 2/a\\
|1 + cot |-||
\ \2//
$$\frac{8 \cot^{3}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{3}}$$
3 sin (a)
$$\sin^{3}{\left(a \right)}$$
3/ pi\
cos |a - --|
\ 2 /
$$\cos^{3}{\left(a - \frac{\pi}{2} \right)}$$
2 / pi\
2*sin (a)*sin|a + --|
\ 2 /
------------------------
/ 2 \
| sin (2*a)|
|1 + ---------|*sin(2*a)
| 4 |
\ 4*sin (a)/
$$\frac{2 \sin^{2}{\left(a \right)} \sin{\left(a + \frac{\pi}{2} \right)}}{\left(1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}\right) \sin{\left(2 a \right)}}$$
/ 1 2 \
|- ------------- + -------------|*sec(a)
| / pi\ / pi\|
| sec|4*a - --| sec|2*a - --||
\ \ 2 / \ 2 //
----------------------------------------
8
$$\frac{\left(- \frac{1}{\sec{\left(4 a - \frac{\pi}{2} \right)}} + \frac{2}{\sec{\left(2 a - \frac{\pi}{2} \right)}}\right) \sec{\left(a \right)}}{8}$$
1
------------
3/ pi\
sec |a - --|
\ 2 /
$$\frac{1}{\sec^{3}{\left(a - \frac{\pi}{2} \right)}}$$
1 -------------------- / 2 \ | csc (a)| |1 + -------|*csc(a) | 2 | \ sec (a)/
$$\frac{1}{\left(\frac{\csc^{2}{\left(a \right)}}{\sec^{2}{\left(a \right)}} + 1\right) \csc{\left(a \right)}}$$
1 ------------------------- / 2 \ | csc (a) | |1 + ------------|*csc(a) | 2/pi \| | csc |-- - a|| \ \2 //
$$\frac{1}{\left(\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(a \right)}}$$
2/a\
-1 + cot |-|
\2/
----------------------------------
/ 2 \ / 2/a\\
\1 + cot (a)/*|1 + cot |-||*cot(a)
\ \2//
$$\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\cot^{2}{\left(a \right)} + 1\right) \cot{\left(a \right)}}$$
/ 1 2 \ /pi \
|- -------- + --------|*csc|-- - a|
\ csc(4*a) csc(2*a)/ \2 /
-----------------------------------
8
$$\frac{\left(- \frac{1}{\csc{\left(4 a \right)}} + \frac{2}{\csc{\left(2 a \right)}}\right) \csc{\left(- a + \frac{\pi}{2} \right)}}{8}$$
3/a\
8*tan |-|
\2/
--------------
3
/ 2/a\\
|1 + tan |-||
\ \2//
$$\frac{8 \tan^{3}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{3}}$$
/ 2/a\\ / 2*cot(2*a) 4*cot(a) \
|1 + cot |-||*|- ------------- + -----------|
\ \2// | 2 2 |
\ 1 + cot (2*a) 1 + cot (a)/
---------------------------------------------
/ 2/a\\
8*|-1 + cot |-||
\ \2//
$$\frac{\left(- \frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} + \frac{4 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1}\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}{8 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)}$$
(1 + cot(a/2)^2)*(-2*cot(2*a)/(1 + cot(2*a)^2) + 4*cot(a)/(1 + cot(a)^2))/(8*(-1 + cot(a/2)^2))