Question

The number of solutions of $2^{\sin |x|}=3^{|\cos x|},x\in [-\pi ,\pi ]$ is equal to

Answer 1

$2^{\sin |x|}=3^{|\cos x|}$
Taking log both sides,
$\frac{\sin |x|}{|\cos x|}=\frac{\log 3}{\log 2}$
$\Rightarrow |\tan x|=\log \frac{3}{2}$ (since $\sin |x|$ and $|\sin x|$ are same in $[-\pi ,\pi ])$
Value of $\tan x$ is positive in $1^{\text{st}}$ and $3^{\text{rd}}$ quadrant and negative in $2^{\text{nd}}$ and $4^{\text{th}}$ quadrant.
$\therefore \tan x$ will repeat its values in all $4$ quadrants.
$\therefore$ Number of solutions $=4$