The number of solutions, of the equation e sin x − 2 e − sin x = 2 is

Let, $e^{\sin x}=t$

$\Rightarrow t-\frac{2}{t}=2$

$\Rightarrow t^2-2t-2=0$

$\Rightarrow t^2-2t+1=3$

$\Rightarrow {\left(t-1\right)}^2=3$

$\Rightarrow t=1\pm \sqrt{3}$

$\Rightarrow t\approx 2.73, -0.73\left(\text{not possible}\right)$

$\Rightarrow t\approx 2.73$

$\Rightarrow e^{\sin x}\approx 2.73$

$\Rightarrow \sin x\approx {\log }_e\left(2.73\right)$

$\Rightarrow \sin x>1\to \text{Not possible}$

So no solutions will be possible for the given equation.