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