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Q.
The number of solutions to the equation $e^{\sin x}-e^{-\sin x}-4=0$ is
Trigonometric Functions
Solution:
Put $e^{\sin x}=t$
$\Rightarrow t^{2}-4 t-1=0$
$\Rightarrow t=e^{\sin x}=2 \pm \sqrt{5}$
$\Rightarrow \sin x=\log _{e}(2+\sqrt{5})$ 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.