Question

The number of solution(s) of the equation $e^{\sin x}-e^{-\sin x}=4,(e\approx 2.72)$ is

• A. $0$
• B. $1$
• C. $2$
• D. $4$

Answer 1

The correct option is A $0$

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

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

Hence, the given equation has no solution.