Question

The number of solutions to the equation $e^{\sin x}-e^{-\sin x}-4=0$ is

• A. 1
• B. 2
• C. 4
• D. 0

Answer 1

Answer: (D)

Put $e^{\sin x}=t$
$\Rightarrow t^2-4t-1=0$
$\Rightarrow t=e^{\sin x}=2\pm \sqrt{5}$
$\Rightarrow \sin x={\log }_e(2+\sqrt{5})$ or ${\log }_e(2-\sqrt{5})$
$2+\sqrt{5}>e\Rightarrow \sin x>1\text{ and }2-\sqrt{5}$
=negative and ${\log }_e$ is defined only for positive values.
Hence, there does not exist any solution.