The number of real roots of the equation

Question

$$e^{\sin x}-e^{-\sin x}-4=0$$

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 ?

Answer 1

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.

Answer 2

$e^{\sin x}-e^{-\sin x}-4 \le e-e^{-\sin x}-4<0.$ Hence, the equation has no real roots !