in order for tan x = 0, sin x must be = 0
sin x = 0 at 0 degrees and 180 degrees
so, tan x = 0 at 0, pi
as for the period being pi, you need to have tan x be negative and positive
so remember the acronym All students take calculus ( ASTC) for where they are positive.
tan x is negative in Quadrant II and positive in Quadrant I, since it just repeats after that, the period therefore must be Pi.