If $\mathrm{f}\left(\mathrm{x}\right)=\sqrt{\cos (\sin \mathrm{x})}+\sqrt{\sin (\cos \mathrm{x})}$ , then the range of $\mathrm{f}\left(\mathrm{x}\right)$ is 

Period of f(x) is $2\mathrm{\pi }$ , but f(x) is not defined for $\mathrm{x}\in (\mathrm{\pi }/2,3\mathrm{\pi }/2)$. Hence it suffices to consider $\mathrm{ }\mathrm{x}\in (-\mathrm{\pi }/2,\mathrm{\pi }/2)$ Further since f(x) is even, we consider $\mathrm{x}\in [0,\mathrm{\pi }/2]$ Now $\sqrt{\cos (\sin \mathrm{x})} \mathrm{and} \sqrt{\sin (\cos \mathrm{x})}$ are decreasing functions for $\mathrm{x}\in [\mathrm{\pi },\mathrm{\pi }/2].\Rightarrow {\mathrm{R}}_{\mathrm{f}}=\left[\mathrm{f}\right(\mathrm{\pi }/2),\mathrm{f}(0\left)\right]=[\sqrt{\cos 1},1+\sqrt{\sin 1}]$