Question

The fundamental period of the function $\frac{|\sin x|+|\cos x|}{|\sin x-\cos x|}$ is

• A. $\frac{\pi }{2}$
• B. $\frac{\pi }{4}$
• C. $\pi$
• D. $2\pi$

Answer 1

The correct option is C $\pi$

The period $T=\text{LCM}\{\text{period of}\sin x,\text{period of}\cos x\}=2\pi$

Whenever there is a trigonometric function involved, after finding $T$, always check whether the $\frac{T}{2}$ is a period or not.

When period $p=\pi$, then
$\frac{|\sin x|+|\cos x|}{|\sin x-\cos x|}=\frac{|-\sin x|+|-\cos x|}{|-\sin x+\cos x|}=\frac{|\sin x|+|\cos x|}{|\sin x-\cos x|}$

When period $p=\frac{\pi }{2}$, then
$\frac{|\sin x|+|\cos x|}{|\sin x-\cos x|}=\frac{|\cos x|+|\sin x|}{|\cos x+\sin x|}$
$\Rightarrow \frac{\pi }{2}$ cannot be the period.

Hence, the fundamental period of the function is $\pi$.