The equation \(\mathrm{e}^{\sin x}-\mathrm{e}^{-\sin x}=4\) has _______ solutions.

\(\mathrm{e}^{\sin x}-e^{-\sin x}-4=0\)
Let \(\mathrm{e}^{\sin x}=y\), then the given equation can be written as \(y^2-4 y-1=0 \Rightarrow y=2 \cdot \pm \sqrt{5}\)
But the value of \(y=\mathrm{e}^{\sin x}\) is always positive, so
\(\begin{aligned}
& y=2+\sqrt{5} \\
& \Rightarrow \mathrm{e}^{\sin x}=2+\sqrt{5} \\
& \Rightarrow \mathrm{e}^{\sin x}\gt\mathrm{e} \\
& \Rightarrow \sin x\gt1
\end{aligned}\)
\(\begin{aligned}
& \ldots[\because 2 \lt \sqrt{5}] \\
& \ldots[\because 2+\sqrt{5}\gt\mathrm{e}]
\end{aligned}\)
which is not possible, since \(\sin x\) cannot be greater than 1.
Hence, no real value of \(x\) satisfies the given equation.