6. Find the set of values of the function $f(x)=2 \sin ((\pi / 4) \sin (\sqrt{x-2}+x+2)-(5 \pi / 2))$.
Solution:

$f(x)=2 \sin \left(\frac{\pi}{4} \sin (\sqrt{x-2}+x+2)-\frac{5 \pi}{2}\right)=-2 \cos \left(\frac{\pi}{4} \sin (\sqrt{x-2}+x+2)\right)$ Since $\sqrt{x-2}+x+2 \in[4 ; \infty)$, then $\sin (\sqrt{x-2}+x+2) \in[-1 ; 1], \quad \frac{\pi}{4} \sin (\sqrt{x-2}+x+2) \in[-\pi / 4 ; \pi / 4]$.

Therefore, $\cos \left(\frac{\pi}{4} \sin (\sqrt{x-2}+x+2)\right) \in[\sqrt{2} / 2 ; 1] \Rightarrow-2 \cos \left(\frac{\pi}{4} \sin (\sqrt{x-2}+x+2)\right) \in[-2 ;-\sqrt{2}]$.

Answer: $[-2 ;-\sqrt{2}]$.