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  4. The period of the function f(x)=(| sin x|-| cos x|/| sin x+ cos x|) is

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

Jharkhand CECEJharkhand CECE 2015

Solution:

We observe that
$ f(x+\pi )=\frac{|\sin (\pi +x)|-|\cos (\pi +x)|}{|\sin (x+\pi )+\cos (x+\pi )|} $
$ \Rightarrow $ $ f(x+\pi )=\frac{|\sin x|-|\cos x|}{|-\sin x-\cos x|} $
$ =\frac{|\sin x|-|\cos x|}{|\sin x+\cos x|} $
$ \Rightarrow $ $ f(x+\pi )=f(x),\,\,\forall x\in R $
Therefore, $ f(x) $ is periodic with period $ \pi $ .