Question

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

• A. $\frac{\pi }{2}$
• B. $2\pi$
• C. $\pi$
• D. None of these

Answer 1

Answer: (C)

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 $\pi$ .