Question

The number of solutions of the equation $e^{\cos x}-e^{-\cos x}=4$ is

Answer 1

Assuming $e^{\cos x}=t$
So we can write
$\Rightarrow t-\frac{1}{t}=4\Rightarrow t^2-4t-1=0\Rightarrow t=\frac{4\pm 2\sqrt{5}}{2}=2\pm \sqrt{5}$
$t=2-\sqrt{5}$ is not possible as $t>0$
As we know that
$-1\le \cos x\le 1\Rightarrow \frac{1}{e}\le t\le e\therefore t\ne 2+\sqrt{5}$

Hence, no solution is possible for the given equation.