Given equation ${\mathrm{e}}^{\sin \mathrm{x}}-{\mathrm{e}}^{- \sin \mathrm{x}}-4=0$

Let ${\mathrm{e}}^{\sin \mathrm{x}}=\mathrm{y},$then given equation can be written as

${\mathrm{y}}^2-4\mathrm{y}-1=0\Rightarrow \mathrm{y}=2\pm \sqrt{5}$

But the value of $\mathrm{y}={\mathrm{e}}^{\sin \mathrm{x}}$ is always positive, so

$y=2+\sqrt{5} (\because 2<\sqrt{5})$

$$\begin{gathered} \Rightarrow {\log }_{\mathrm{e}}\mathrm{y}={\log }_{\mathrm{e}}(2+\sqrt{5})\Rightarrow \sin \mathrm{x} \\ ={\log }_{\mathrm{e}}(2+\sqrt{5})>1 \end{gathered}$$

Which is impossible, since sin x cannot be greater than 1. Hence we cannot find any real value of x which satisfe the given equation