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)
Combinatoria
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-cos(a)*tan(a)
---------------
2
1 + cot (a)
$$- \frac{\cos{\left(a \right)} \tan{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1}$$
-cos(a)*tan(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 /
$$\frac{i \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right) \left(e^{i a} - e^{- i a}\right)}{\left(e^{i a} + e^{- i a}\right) \left(\cot^{2}{\left(a \right)} + 1\right)}$$
-cos(a)*tan(a)
---------------
2
1 + cot (a)
$$- \frac{\cos{\left(a \right)} \tan{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1}$$
-cos(a)*tan(a)/(1 + cot(a)^2)
Denominador racional
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-cos(a)*tan(a)
---------------
2
1 + cot (-a)
$$- \frac{\cos{\left(a \right)} \tan{\left(a \right)}}{\cot^{2}{\left(- a \right)} + 1}$$
-cos(a)*tan(a)/(1 + cot(-a)^2)
Parte trigonométrica
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3/ pi\
-cos |a - --|
\ 2 /
$$- \cos^{3}{\left(a - \frac{\pi}{2} \right)}$$
-sin(a)
-----------
2
cos (a)
1 + -------
2
sin (a)
$$- \frac{\sin{\left(a \right)}}{1 + \frac{\cos^{2}{\left(a \right)}}{\sin^{2}{\left(a \right)}}}$$
-1 ------------------------------ / 2/ pi\\ | sec |a - --|| | \ 2 /| / pi\ |1 + ------------|*sec|a - --| | 2 | \ 2 / \ sec (a) /
$$- \frac{1}{\left(1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}\right) \sec{\left(a - \frac{\pi}{2} \right)}}$$
3/a\
-8*tan |-|
\2/
--------------
3
/ 2/a\\
|1 + tan |-||
\ \2//
$$- \frac{8 \tan^{3}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{3}}$$
-1 ------- 3 csc (a)
$$- \frac{1}{\csc^{3}{\left(a \right)}}$$
-1
------------
3/ pi\
sec |a - --|
\ 2 /
$$- \frac{1}{\sec^{3}{\left(a - \frac{\pi}{2} \right)}}$$
/ / 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)
$$- \frac{2 \sin{\left(2 a \right)} - \sin{\left(4 a \right)}}{8 \cos{\left(a \right)}}$$
3/a\
-8*cot |-|
\2/
--------------
3
/ 2/a\\
|1 + cot |-||
\ \2//
$$- \frac{8 \cot^{3}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{3}}$$
-(-sin(4*a) + 2*sin(2*a))
--------------------------
/ pi\
8*sin|a + --|
\ 2 /
$$- \frac{2 \sin{\left(2 a \right)} - \sin{\left(4 a \right)}}{8 \sin{\left(a + \frac{\pi}{2} \right)}}$$
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 \ | 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\\ / 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//
$$- \frac{\left(- \frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} + \frac{4 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1}\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}{8 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)}$$
/ 2/a\\
-|1 - tan |-||*tan(a)
\ \2//
---------------------------
/ 1 \ / 2/a\\
|1 + -------|*|1 + tan |-||
| 2 | \ \2//
\ tan (a)/
$$- \frac{\left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right) \tan{\left(a \right)}}{\left(1 + \frac{1}{\tan^{2}{\left(a \right)}}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
3 -sin (a)
$$- \sin^{3}{\left(a \right)}$$
/ 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 \
-|- ------------- + -------------|*sec(a)
| / pi\ / pi\|
| sec|4*a - --| sec|2*a - --||
\ \ 2 / \ 2 //
------------------------------------------
8
$$- \frac{\left(- \frac{1}{\sec{\left(4 a - \frac{\pi}{2} \right)}} + \frac{2}{\sec{\left(2 a - \frac{\pi}{2} \right)}}\right) \sec{\left(a \right)}}{8}$$
2 -2*sin (a)*cos(a) ------------------------ / 2 \ | sin (2*a)| |1 + ---------|*sin(2*a) | 4 | \ 4*sin (a)/
$$- \frac{2 \sin^{2}{\left(a \right)} \cos{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}\right) \sin{\left(2 a \right)}}$$
/ pi\
cos|-a - --|
\ 2 /
-----------------
2
cos (a)
1 + -------------
2/ pi\
cos |-a - --|
\ 2 /
$$\frac{\cos{\left(- a - \frac{\pi}{2} \right)}}{\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(- a - \frac{\pi}{2} \right)}} + 1}$$
/ 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//
$$- \frac{\left(- \frac{2 \tan{\left(2 a \right)}}{\tan^{2}{\left(2 a \right)} + 1} + \frac{4 \tan{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}{8 \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)}$$
-1 -------------------- / 2 \ | csc (a)| |1 + -------|*csc(a) | 2 | \ sec (a)/
$$- \frac{1}{\left(\frac{\csc^{2}{\left(a \right)}}{\sec^{2}{\left(a \right)}} + 1\right) \csc{\left(a \right)}}$$
3*sin(a) sin(3*a)
- -------- + --------
4 4
$$- \frac{3 \sin{\left(a \right)}}{4} + \frac{\sin{\left(3 a \right)}}{4}$$
/ 1 2 \ /pi \
-|- -------- + --------|*csc|-- - a|
\ csc(4*a) csc(2*a)/ \2 /
-------------------------------------
8
$$- \frac{\left(- \frac{1}{\csc{\left(4 a \right)}} + \frac{2}{\csc{\left(2 a \right)}}\right) \csc{\left(- a + \frac{\pi}{2} \right)}}{8}$$
-(-1/csc(4*a) + 2/csc(2*a))*csc(pi/2 - a)/8
Unión de expresiones racionales
[src]
-cos(a)*tan(a)
---------------
2
1 + cot (a)
$$- \frac{\cos{\left(a \right)} \tan{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1}$$
-cos(a)*tan(a)/(1 + cot(a)^2)