In the interval $\left[–\mathrm{\pi },\mathrm{ }\mathrm{\pi }\right]$ all the values of $\sin \left|x\right|$ are positive as well as $\left|\cos x\right|$
Hence in $\left[–\mathrm{\pi },\mathrm{ }\mathrm{\pi }\right]$, equation gets reduced to, 

$2^{\sin x}=3^{\cos x}$
Taking log on both sides,
$\sin x\log 2=\cos x\log 3$
$\Rightarrow \left|\tan x\right|=\log 3/\log 2$ , value of $\tan x$ are positive in $1^{st}$ and $3^{rd}$ quadrant and negative in $2^{nd}$ and $4^{th}$ quadrant but positive for $\left|\tan x\right|$ in all the values of $x$ in $\left[–\mathrm{\pi },\mathrm{ }\mathrm{\pi }\right]$.

Hence, $\tan x$ will repeat its value in all $4$ quadrants, so the number of solutions are $4$ .