[tex]\tan x=0[/tex] for [tex]x=n\pi[/tex], where [tex]n[/tex] is any integer. The inverse tangent function returns numbers between [tex]-\dfrac\pi2[/tex] and [tex]\dfrac\pi2[/tex]. The only multiple of [tex]\pi[/tex] in this range is 0, so [tex]\tan^{-1}0=0[/tex].
Then
[tex]\sin\left(\tan^{-1}0\right)=\sin0=\boxed0[/tex]
