The domain of the function

$f\left(x\right)=\frac{1}{\sqrt{\left\{\sin x\right\}+\left\{\sin \left(\pi +x\right)\right\}}}$ where $\left\{ \right\}$ denotes the fractional part, is

$f\left(x\right)=\frac{1}{\sqrt{\left\{\sin x\right\}+\left\{\sin \left(\pi +x\right)\right\}}}$

$=\frac{1}{\sqrt{\left\{\sin x\right\}+\left\{-\sin x\right\}}}$

Now

$\left\{\sin x\right\}+\left\{-\sin x\right\}=\left\{\begin{array}{c}\text{0\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}sinx\hspace{0.17em}is\hspace{0.33em}an\hspace{0.33em}integer}\\ 1\sin x\text{\hspace{0.33em}is\hspace{0.33em}not\hspace{0.33em}an\hspace{0.33em}integer}\end{array}\right.$

For f(x) to get defined

$\left\{\sin x\right\}+\left\{-\sin x\right\}\ne 0$

$\Rightarrow \sin x\ne$integer

$\Rightarrow \sin x\ne \pm 1,0\Rightarrow x\ne \frac{n\pi }{2},n\in I$

Hence, the domain is $R-\left\{\frac{n\pi }{2}/n\in I\right\}$