Question

The number of solutions of the equation $e^{\sin x}-e^{-\sin x}-4=0,(e\simeq 2.73)$ is

• A. $1$
• B. $2$
• C. $3$
• D. no solution

Answer 1

The correct option is D no solution

$e^{\sin x}-\frac{1}{e^{\sin x}}-4=0\Rightarrow (e^{\sin x}{)}^2-4e^{\sin x}+4=5\Rightarrow (e^{\sin x}{)}^2-4e^{\sin x}-1=0\Rightarrow e^{\sin x}=\frac{4\pm \sqrt{20}}{2}\Rightarrow e^{\sin x}=2\pm \sqrt{5}$

Now,
$\sin x\in [-1,1]$
$\Rightarrow e^{\sin x}\in [e^{-1},e^1]2\pm \sqrt{5}\notin [e^{-1},e^1]$

Hence, there does not exist any solution.