Range of: $\sqrt{\sin(\cos x)}+\sqrt{\cos(\sin x)}$ Range of:
$$\sqrt{\sin(\cos x)}+\sqrt{\cos(\sin x)}$$
Any help will be appreciated.

The given expression $f(x)$ is even and $2\pi$-periodic. Therefore it is enough to consider the interval $[0,\pi]$. 
When ${\pi\over2}<x\leq\pi$ then $0>\cos x\geq-1>-{\pi\over2}$; so $\sqrt{\sin(\cos x)}$ is not defined in this interval. 
When $0\leq x\leq{\pi\over2}$ then both $\cos x$ and $\sin x$ are in $[0,1]\subset\bigl[0,{\pi\over2}\bigr]$. It follows that in this $x$-interval both compositions $\sqrt{\phantom{i}} \circ\sin\circ\cos\ $ and $\ \sqrt{\phantom{i}}\circ\cos\circ\sin$ are decreasing, and so is  $f(x)$. 
Therefore the range of $f(x)$ is the interval $\left[f\bigl({\pi\over2}\bigr), f(0)\right]=\bigl[\sqrt{\cos 1}, 1+\sqrt{\sin 1}\bigr]$.
