So when tan(x) < 1, that means that cos(x) > sin(x) and the opposite is true when tan(x) > 1. They are clearly equal when it equals 1.
Find where tan(x)=1 and then draw out the graph by hand a couple times to see the pattern. It is periodic. As for domain, what x-values are excluded from these functions? As for range, the lowest point for g(x) is when the biggest difference is between sin(x) and cos(x). The highest point is when they intersect, because at any point before and after the intersection one of the two must be below the other.