$\text{The domain of }f\left(x\right)=\sqrt{\cos \left(\sin x\right)}$is

$f\left(x\right)=\sqrt{\cos (\sin x)}\text{ is defined if }$$\cos (\sin x)\ge 0----\left(1\right)$

$\text{ As we know, }-1\le \sin x\le 1\text{ for all }x\in R\text{ . So, }$$\cos \theta \ge 0$

$\text{ (Here, }\theta =\sin x\text{ lies in the first and fourth quadrants) }$

$\text{ i.e., }\cos (\sin x)\ge 0\text{ for all }x$

$\text{ i.e., }x\in R$

$\text{ Thus, the domain of }f\left(x\right)\text{ is }R\text{ . }$