The period of the function f(x) = frac{|sin x | vedclass

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Question

(A) Let $f(x) = \frac{|\sin x| + |\cos x|}{|\sin x - \cos x|}$.Square the function to simplify: $f(x)^2 = \frac{(|\sin x| + |\cos x|)^2}{(|\sin x - \c

Vedclass · Accepted Answer

$\pi / 2$

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(A) Let $f(x) = \frac{|\sin x| + |\cos x|}{|\sin x - \cos x|}$.Square the function to simplify: $f(x)^2 = \frac{(|\sin x| + |\cos x|)^2}{(|\sin x - \cos x|)^2} = \frac{\sin^2 x + \cos^2 x + 2|\sin x \cos x|}{\sin^2 x + \cos^2 x - 2\sin x \cos x} = \frac{1 + |\sin 2x|}{1 - \sin 2x}$.Alternatively,observe the symmetry: $f(x + \pi/2) = \frac{|\sin(x + \pi/2)| + |\cos(x + \pi/2)|}{|\sin(x + \pi/2) - \cos(x + \pi/2)|} = \frac{|\cos x| + |\sin x|}{|\cos x - (-\sin x)|} = \frac{|\sin x| + |\cos x|}{|\cos x + \sin x|} = f(x)$.Since $f(x + \pi/2) = f(x)$,the period is $\pi/2$.

The period of the function $f(x) = \frac{|\sin x| + |\cos x|}{|\sin x - \cos x|}$ is

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