Why is tan (0) = 0?

You may see this from either the definition of ##tan## or from geometry.
1] the tangent is equal to the ratio ##sin/cos## so if you have:
##tan(0)=sin(0)/cos(0)=0/1=0##
2] obseving the geometrical meaning of the tangent you have:
You can see that when ##alpha## becomes very small or zero the length of the red segment representing the ##tan## becomes small and eventually zero when ##alpha=0##.
Hope it helps!