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.

$$u=\cos(x); v=\sin(x)$$
$$\sqrt{\sin(u)}+\sqrt{\cos(v)}$$
$\sin(u)$ must be greater than or equal to 0, and $\cos(v)$ must be greater than or equal to 0:
$$u\in{[2\pi{k},\pi+2\pi{k}],k\in\mathbb{Z}}$$
$$v\in{[-\frac{\pi}{2}+2\pi{k},\frac{\pi}{2}+2\pi{k}]},k\in\mathbb{Z}$$
Since $v=\sin{x}$, $v\in{[-1,1]}$, $v$ will always be in the necessary interval. But $u$ will only be in the necessary interval when $\cos{x}\ge0$. That is, when $x\in[-\frac{\pi}{2}+2\pi{k},\frac{\pi}{2}+2\pi{k}]\bigcap[2\pi{k},\pi+2\pi{k}]$.
So the domain is $[2\pi{k},\frac{\pi}{2}+2\pi{k}],k\in\mathbb{Z}$. Now find the set of function values over that domain. Note that the function is not strictly increasing or decreasing over the domain, so the minimum and maximum are not the endpoints of the domain.
