Question

The period of $f\left(x\right)=\frac{1}{2}\left(\frac{\left|\sin x\right|}{\cos x}+\frac{\sin x}{\left|\cos x\right|}\right)$ is

• A. $-\mathrm{\pi }$
• B. $2\mathrm{\pi }$
• C. $\frac{\mathrm{\pi }}{2}$
• D. $\frac{\mathrm{\pi }}{3}$

Answer 1

The correct option is B

$2\mathrm{\pi }$

Explanation for the correct option:

Step 1: Determine the period of $f\left(x\right)$:

The given function is,

$f\left(x\right)=\frac{1}{2}\left[\frac{\left|\sin x\right|}{\cos x}+\frac{\sin x}{\left|\cos x\right|}\right]$ $...\left(\mathrm{i}\right)$

Replacing $x$ by $\left(2\mathrm{\pi }+\mathrm{x}\right)$ in the above,

$f\left(2\mathrm{\pi }+x\right)=\frac{1}{2}\left[\frac{\left|\sin \left(2\mathrm{\pi }+x\right)\right|}{\cos \left(2\mathrm{\pi }+x\right)}+\frac{\sin \left(2\mathrm{\pi }+x\right)}{\left|\cos \left(2\mathrm{\pi }+x\right)\right|}\right]$

$\Rightarrow f\left(2\mathrm{\pi }+x\right)=\frac{1}{2}\left[\frac{\left|\sin x\right|}{\cos x}+\frac{\mathrm{sinx}}{\left|\cos x\right|}\right]$ $[\because \sin (2\pi +x)=\mathrm{sinx},\cos (2\pi +x)=\mathrm{cosx}]$

$\Rightarrow f\left(2\mathrm{\pi }+x\right)=f\left(x\right)$ [using equation $\left(\mathrm{i}\right)$]

So, the period of $f\left(x\right)$ is $2\mathrm{\pi }$.

Step 2: Check for the minimum period:

Again, replacing $x$ by $\left(\mathrm{\pi }+\mathrm{x}\right)$ in the equation $\left(\mathrm{i}\right)$,

$f\left(\mathrm{\pi }+x\right)=\frac{1}{2}\left[\frac{\left|\sin \left(\mathrm{\pi }+x\right)\right|}{\cos \left(\mathrm{\pi }+x\right)}+\frac{\sin \left(\mathrm{\pi }+x\right)}{\left|\cos \left(\mathrm{\pi }+x\right)\right|}\right]$

$\Rightarrow f\left(\mathrm{\pi }+x\right)=\frac{1}{2}\left[\frac{\left|-\mathrm{sinx}\right|}{\left\{-\cos x\right\}}+\frac{\left\{-\mathrm{sinx}\right\}}{\left|-\cos x\right|}\right]$ $[\because \sin (\pi +x)=-\mathrm{sinx},\cos (\pi +x)=-\mathrm{cosx}]$

$\Rightarrow f\left(\mathrm{\pi }+x\right)=\frac{1}{2}\left[-\frac{\left|\mathrm{sinx}\right|}{\cos x}-\frac{\mathrm{sinx}}{\left|\cos x\right|}\right]$ $[\because |\pm a|=\left|a\right|]$

$\Rightarrow f\left(\mathrm{\pi }+x\right)=-\frac{1}{2}\left[\frac{\left|\mathrm{sinx}\right|}{\cos x}+\frac{\mathrm{sinx}}{\left|\cos x\right|}\right]$

$\Rightarrow f\left(\mathrm{\pi }+x\right)=-f\left(x\right)$ [using equation $\left(\mathrm{i}\right)$]

So, the period of above is $\mathrm{\pi }$.

Thus, the period of $f\left(x\right)=LCM\left(2\mathrm{\pi },\mathrm{\pi }\right)$

$\Rightarrow$ the period of $f\left(x\right)=2\mathrm{\pi }$

Hence, option $B$ is the correct option.