The number of real roots of the equation 
$$e^{\sin x}-e^{-\sin x}-4=0$$

Let $e^{\sin x}=y$
Then $$y-\frac 1y -4=0$$
$$y^2-4y-1=0$$
$$y=2+\sqrt 5 , 2-\sqrt 5$$
How should I solve further ?

Your method is fine, now observe that


*

*$e^{\sin x}=2+\sqrt 5 \implies \sin  x=\log(2+\sqrt 5)>1$
which is not possible and


*

*$e^{\sin x}=2-\sqrt 5<0 $
which is not possible, therefore there are not real solutions.
