Question

The period of the function
$f\left(x\right)=2^{\sqrt{\frac{2}{1-\cos 2x}}}+4^{\sqrt{\frac{1+{\tan }^{2}x}{4}}}$

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

Answer 1

The correct option is A $2\pi$

$f\left(x\right)=2^{\sqrt{\frac{2}{1-\cos 2x}}}+4^{\sqrt{\frac{1+{\tan }^{2}x}{4}}}$

$=2^{\sqrt{\frac{2}{1+2{\sin }^{2}x-1}}}+4^{\sqrt{\frac{{\sec }^{2}x}{4}}}$

$=2^{\sqrt{\frac{1}{{\sin }^{2}x}}}+2^{2\times \sqrt{\frac{{\sec }^{2}x}{4}}}$

$=2^{cosecx}+2^{secx}$

In case of $cosecx$ and $secx$

Period $=2\pi$

Domain: $-\infty <x<+\infty ,x\ne (2n+1)\pi )/2,n\epsilon Z$