The function sinh(sinx) is increasing is

$$\begin{gathered} \text{⨍ (x)=}\frac{{\mathrm{e}}^{\sin \mathrm{x}} -{\mathrm{e}}^{-\sin \mathrm{x}}}{2} \\ {\text{⨍ }}^1\text{(x)=}\frac{{\mathrm{e}}^{\sin \mathrm{x}} \cos \mathrm{x}-{\mathrm{e}}^{-\sin \mathrm{x}}(-\cos \mathrm{x})}{ 2} \\ =\cos \mathrm{x}\left(\frac{{\mathrm{e}}^{\sin \mathrm{x}}+{\mathrm{e}}^{-\sin x}}{2}\right) \end{gathered}$$

$\text{⨍ (x)}$ is increasing

$$\begin{gathered} \Rightarrow {\text{⨍ }}^1\text{(x)>0} \\ \cos \mathrm{x}>0 \\ \mathrm{x}\in \left(2\mathrm{n\pi }-\frac{\mathrm{\pi }}{2}, 2\mathrm{n\pi }+\frac{\mathrm{\pi }}{2}\right) \end{gathered}$$