$Thedomainoff\left(x\right)=\frac{1}{\left|\sin x\right|+\sin x}is$

$\begin{array}{l}f\left(x\right)=\frac{1}{\left|\sin x\right|+\sin x}\\ isdefinediff\left|\sin x\right|+\sin x\ne 0\\ \left|\sin x\right|+\sin x=\{\begin{array}{l}2\sin xif2n\pi <x<(2n+1)\pi \\ 0Otherwise\end{array}\\ \therefore Domain=\underset{n\in Z}{\cup }(2n\pi ,(2n+1)\pi )\end{array}$