Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
(cos(x)/(1+sin(x)))+(cos(x)/(1-sin(x)))
¿Cómo vas a descomponer esta (cos(x)/(1+sin(x)))+(cos(x)/(1-sin(x))) expresión en fracciones?
Solución
Ha introducido
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cos(x) cos(x) ---------- + ---------- 1 + sin(x) 1 - sin(x)
$$\frac{\cos{\left(x \right)}}{\sin{\left(x \right)} + 1} + \frac{\cos{\left(x \right)}}{1 - \sin{\left(x \right)}}$$
cos(x)/(1 + sin(x)) + cos(x)/(1 - sin(x))
Potencias
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I*x -I*x I*x -I*x
e e e e
---- + ----- ---- + -----
2 2 2 2
---------------------- + ----------------------
/ -I*x I*x\ / -I*x I*x\
I*\- e + e / I*\- e + e /
1 + ------------------ 1 - ------------------
2 2
$$\frac{\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}}{\frac{i \left(e^{i x} - e^{- i x}\right)}{2} + 1} + \frac{\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}}{- \frac{i \left(e^{i x} - e^{- i x}\right)}{2} + 1}$$
(exp(i*x)/2 + exp(-i*x)/2)/(1 + i*(-exp(-i*x) + exp(i*x))/2) + (exp(i*x)/2 + exp(-i*x)/2)/(1 - i*(-exp(-i*x) + exp(i*x))/2)
Combinatoria
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-2*cos(x) -------------------------- (1 + sin(x))*(-1 + sin(x))
$$- \frac{2 \cos{\left(x \right)}}{\left(\sin{\left(x \right)} - 1\right) \left(\sin{\left(x \right)} + 1\right)}$$
-2*cos(x)/((1 + sin(x))*(-1 + sin(x)))
Denominador común
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-2*cos(x)
------------
2
-1 + sin (x)
$$- \frac{2 \cos{\left(x \right)}}{\sin^{2}{\left(x \right)} - 1}$$
-2*cos(x)/(-1 + sin(x)^2)
Denominador racional
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(1 - sin(x))*cos(x) + (1 + sin(x))*cos(x)
-----------------------------------------
(1 - sin(x))*(1 + sin(x))
$$\frac{\left(1 - \sin{\left(x \right)}\right) \cos{\left(x \right)} + \left(\sin{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\left(1 - \sin{\left(x \right)}\right) \left(\sin{\left(x \right)} + 1\right)}$$
((1 - sin(x))*cos(x) + (1 + sin(x))*cos(x))/((1 - sin(x))*(1 + sin(x)))
Unión de expresiones racionales
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2*cos(x) ------------------------- (1 - sin(x))*(1 + sin(x))
$$\frac{2 \cos{\left(x \right)}}{\left(1 - \sin{\left(x \right)}\right) \left(\sin{\left(x \right)} + 1\right)}$$
2*cos(x)/((1 - sin(x))*(1 + sin(x)))
Respuesta numérica
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cos(x)/(1.0 - sin(x)) + cos(x)/(1.0 + sin(x))
cos(x)/(1.0 - sin(x)) + cos(x)/(1.0 + sin(x))
Parte trigonométrica
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4 ------------------------------- / 1 \ /pi \ |1 + -------------|*csc|-- - x| | /pi \| \2 / | csc|-- - 2*x|| \ \2 //
$$\frac{4}{\left(1 + \frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}\right) \csc{\left(- x + \frac{\pi}{2} \right)}}$$
/pi \
2*csc|-- - x|
\2 /
$$2 \csc{\left(- x + \frac{\pi}{2} \right)}$$
/ 2/x\\
2*|1 - 2*sin |-||
\ \2//
-----------------
2
cos (x)
$$\frac{2 \left(1 - 2 \sin^{2}{\left(\frac{x}{2} \right)}\right)}{\cos^{2}{\left(x \right)}}$$
/ 2/x\\
4*|-1 + cot |-||
\ \2//
--------------------------------
/ 2 \
/ 2/x\\ | -1 + cot (x)|
|1 + cot |-||*|1 + ------------|
\ \2// | 2 |
\ 1 + cot (x) /
$$\frac{4 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\left(\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
1 1 ------------------- + ------------------- / 1 \ / 1 \ |1 + ------|*sec(x) |1 - ------|*sec(x) \ csc(x)/ \ csc(x)/
$$\frac{1}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right) \sec{\left(x \right)}} + \frac{1}{\left(1 - \frac{1}{\csc{\left(x \right)}}\right) \sec{\left(x \right)}}$$
2 ----------- / pi\ sin|x + --| \ 2 /
$$\frac{2}{\sin{\left(x + \frac{\pi}{2} \right)}}$$
2/x\ 2/x\
-1 + cot |-| -1 + cot |-|
\2/ \2/
------------------------------- + -------------------------------
/ /x\ \ / /x\ \
| 2*cot|-| | | 2*cot|-| |
/ 2/x\\ | \2/ | / 2/x\\ | \2/ |
|1 + cot |-||*|1 - -----------| |1 + cot |-||*|1 + -----------|
\ \2// | 2/x\| \ \2// | 2/x\|
| 1 + cot |-|| | 1 + cot |-||
\ \2// \ \2//
$$\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\left(1 - \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
/ pi\
4*sin|x + --|
\ 2 /
-----------------
/pi \
1 + sin|-- + 2*x|
\2 /
$$\frac{4 \sin{\left(x + \frac{\pi}{2} \right)}}{\sin{\left(2 x + \frac{\pi}{2} \right)} + 1}$$
cos(x) cos(x)
--------------- + ---------------
/ pi\ / pi\
1 - cos|x - --| 1 + cos|x - --|
\ 2 / \ 2 /
$$\frac{\cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)} + 1} + \frac{\cos{\left(x \right)}}{1 - \cos{\left(x - \frac{\pi}{2} \right)}}$$
/ 2/x\\
2*|1 + tan |-||
\ \2//
---------------
2/x\
1 - tan |-|
\2/
$$\frac{2 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}{1 - \tan^{2}{\left(\frac{x}{2} \right)}}$$
/ 2/x\\
2*|1 + cot |-||
\ \2//
---------------
2/x\
-1 + cot |-|
\2/
$$\frac{2 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}{\cot^{2}{\left(\frac{x}{2} \right)} - 1}$$
2/x\ 2/x\
1 - tan |-| 1 - tan |-|
\2/ \2/
------------------------------- + -------------------------------
/ /x\ \ / /x\ \
| 2*tan|-| | | 2*tan|-| |
/ 2/x\\ | \2/ | / 2/x\\ | \2/ |
|1 + tan |-||*|1 - -----------| |1 + tan |-||*|1 + -----------|
\ \2// | 2/x\| \ \2// | 2/x\|
| 1 + tan |-|| | 1 + tan |-||
\ \2// \ \2//
$$\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\left(1 - \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
4*cos(x) ------------ 1 + cos(2*x)
$$\frac{4 \cos{\left(x \right)}}{\cos{\left(2 x \right)} + 1}$$
/ pi\ / pi\ sin|x + --| sin|x + --| \ 2 / \ 2 / ----------- + ----------- 1 - sin(x) 1 + sin(x)
$$\frac{\sin{\left(x + \frac{\pi}{2} \right)}}{\sin{\left(x \right)} + 1} + \frac{\sin{\left(x + \frac{\pi}{2} \right)}}{1 - \sin{\left(x \right)}}$$
/ 2/x\\
4*|1 - tan |-||
\ \2//
-------------------------------
/ 2 \
/ 2/x\\ | 1 - tan (x)|
|1 + tan |-||*|1 + -----------|
\ \2// | 2 |
\ 1 + tan (x)/
$$\frac{4 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)}{\left(\frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
4 --------------------- / 1 \ |1 + --------|*sec(x) \ sec(2*x)/
$$\frac{4}{\left(1 + \frac{1}{\sec{\left(2 x \right)}}\right) \sec{\left(x \right)}}$$
1 1 ------------------------ + ------------------------ / 1 \ / 1 \ |1 + -----------|*sec(x) |1 - -----------|*sec(x) | / pi\| | / pi\| | sec|x - --|| | sec|x - --|| \ \ 2 // \ \ 2 //
$$\frac{1}{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \sec{\left(x \right)}} + \frac{1}{\left(1 - \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \sec{\left(x \right)}}$$
2 ------ cos(x)
$$\frac{2}{\cos{\left(x \right)}}$$
1 1 ------------------------ + ------------------------ / 1 \ /pi \ / 1 \ /pi \ |1 + ------|*csc|-- - x| |1 - ------|*csc|-- - x| \ csc(x)/ \2 / \ csc(x)/ \2 /
$$\frac{1}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\left(1 - \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(- x + \frac{\pi}{2} \right)}}$$
2*sec(x)
$$2 \sec{\left(x \right)}$$
2*sec(x)