Question

The period of following function: $f\left(x\right)=\frac{1}{2}\{\frac{|\sin x|}{\cos x}+\frac{|\cos x|}{\sin x}\}$ is $z\pi$. Find $z$.

Answer 1

Let
$f\left(x\right)=\frac{|sinx|}{cosx}+\frac{|cosx|}{sinx}$
Now
$|sin\left(x\right)|=|sin(n\pi +x)|$. similarly $|cosx|=|cos(n\pi +x)|$
Hence the periodicity will depend on the denominators $cosx$ and $sinx$.
Now
$cos\left(x\right)=cos(-x)=cos(2\pi +x)$ ...(i)
And
$sin\left(x\right)=sin(\pi -x)=sin(2\pi +x)$....(ii)
from i and ii , the common period is $2\pi$.
Thus
$f(2\pi +x)=\frac{|sin(2\pi +x)|}{cos(2\pi +x)}+\frac{|cos(2\pi +x)|}{sin(2\pi +x)}$
$=\frac{|sinx|}{cosx}+\frac{|cosx|}{sinx}$
$=f\left(x\right)$
Hence the period of
$h\left(x\right)=\frac{1}{2}(\frac{|sinx|}{cosx}+\frac{|cosx|}{sinx})$ is $2\pi$.