Sr Examen
¿Cómo vas a descomponer esta cos^2x+1/(1+ctg^2(x))-1 expresión en fracciones?
Solución
Ha introducido
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2 1
cos (x) + ----------- - 1
2
1 + cot (x)
$$\left(\cos^{2}{\left(x \right)} + \frac{1}{\cot^{2}{\left(x \right)} + 1}\right) - 1$$
cos(x)^2 + 1/(1 + cot(x)^2) - 1
Denominador común
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1 2
-1 + ----------- + cos (x)
2
1 + cot (x)
$$\cos^{2}{\left(x \right)} - 1 + \frac{1}{\cot^{2}{\left(x \right)} + 1}$$
-1 + 1/(1 + cot(x)^2) + cos(x)^2
Denominador racional
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2 2 / 2 \
- cot (x) + cos (x)*\1 + cot (x)/
---------------------------------
2
1 + cot (x)
$$\frac{\left(\cot^{2}{\left(x \right)} + 1\right) \cos^{2}{\left(x \right)} - \cot^{2}{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}$$
(-cot(x)^2 + cos(x)^2*(1 + cot(x)^2))/(1 + cot(x)^2)
Unión de expresiones racionales
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2 2 / 2 \
- cot (x) + cos (x)*\1 + cot (x)/
---------------------------------
2
1 + cot (x)
$$\frac{\left(\cot^{2}{\left(x \right)} + 1\right) \cos^{2}{\left(x \right)} - \cot^{2}{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}$$
(-cot(x)^2 + cos(x)^2*(1 + cot(x)^2))/(1 + cot(x)^2)
Combinatoria
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2 2 2 2
cos (x) - cot (x) + cos (x)*cot (x)
-----------------------------------
2
1 + cot (x)
$$\frac{\cos^{2}{\left(x \right)} \cot^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} - \cot^{2}{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}$$
(cos(x)^2 - cot(x)^2 + cos(x)^2*cot(x)^2)/(1 + cot(x)^2)
Potencias
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1 2
-1 + ----------- + cos (x)
2
1 + cot (x)
$$\cos^{2}{\left(x \right)} - 1 + \frac{1}{\cot^{2}{\left(x \right)} + 1}$$
2
/ I*x -I*x\
1 |e e |
-1 + ----------- + |---- + -----|
2 \ 2 2 /
1 + cot (x)
$$\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2} - 1 + \frac{1}{\cot^{2}{\left(x \right)} + 1}$$
-1 + 1/(1 + cot(x)^2) + (exp(i*x)/2 + exp(-i*x)/2)^2
Parte trigonométrica
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1 1
-1 + ---------------- + ------------
2 2/pi \
csc (x) csc |-- - x|
1 + ------------ \2 /
2/pi \
csc |-- - x|
\2 /
$$-1 + \frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\frac{\csc^{2}{\left(x \right)}}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + 1}$$
2
/ 2/x\\
|1 - tan |-||
1 \ \2//
-1 + ----------- + --------------
1 2
1 + ------- / 2/x\\
2 |1 + tan |-||
tan (x) \ \2//
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} - 1 + \frac{1}{1 + \frac{1}{\tan^{2}{\left(x \right)}}}$$
1 2
-1 + ---------------- + cos (x)
2
cos (x)
1 + ------------
2/ pi\
cos |x - --|
\ 2 /
$$\cos^{2}{\left(x \right)} - 1 + \frac{1}{\frac{\cos^{2}{\left(x \right)}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} + 1}$$
1 2/ pi\
-1 + ------------- + sin |x + --|
2 \ 2 /
sin (2*x)
1 + ---------
4
4*sin (x)
$$\sin^{2}{\left(x + \frac{\pi}{2} \right)} - 1 + \frac{1}{1 + \frac{\sin^{2}{\left(2 x \right)}}{4 \sin^{4}{\left(x \right)}}}$$
1 1
-1 + ---------------- + -------
2/ pi\ 2
sec |x - --| sec (x)
\ 2 /
1 + ------------
2
sec (x)
$$-1 + \frac{1}{\sec^{2}{\left(x \right)}} + \frac{1}{1 + \frac{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(x \right)}}}$$
1 2
-1 + ------------- + cos (x)
2
sin (2*x)
1 + ---------
4
4*sin (x)
$$\cos^{2}{\left(x \right)} - 1 + \frac{1}{1 + \frac{\sin^{2}{\left(2 x \right)}}{4 \sin^{4}{\left(x \right)}}}$$
0
$$0$$
1 2
-1 + ----------- + cos (x)
2
cos (x)
1 + -------
2
sin (x)
$$\cos^{2}{\left(x \right)} - 1 + \frac{1}{1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}}$$
2
/ 2/x\\
|-1 + cot |-||
1 \ \2//
-1 + ----------- + ---------------
2 2
1 + cot (x) / 2/x\\
|1 + cot |-||
\ \2//
$$\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} - 1 + \frac{1}{\cot^{2}{\left(x \right)} + 1}$$
1 1
-1 + ----------- + -------
2 2
csc (x) sec (x)
1 + -------
2
sec (x)
$$-1 + \frac{1}{\sec^{2}{\left(x \right)}} + \frac{1}{\frac{\csc^{2}{\left(x \right)}}{\sec^{2}{\left(x \right)}} + 1}$$
-1 + 1/(1 + csc(x)^2/sec(x)^2) + sec(x)^(-2)
Compilar la expresión
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1 2
-1 + ----------- + cos (x)
2
1 + cot (x)
$$\cos^{2}{\left(x \right)} - 1 + \frac{1}{\cot^{2}{\left(x \right)} + 1}$$
-1 + 1/(1 + cot(x)^2) + cos(x)^2