Question

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

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

Answer 1

Answer: (B)

$2\pi$

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

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

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

$⟹f(x+2\pi )=f\left(x\right)⟹2\pi$ is the period of $f\left(x\right)$