Annihilator is a smooth submanifold of the cotangent bundle

Question

Let $X$ be a smooth manifold and let $Y$ be a smooth submanifold. Denote by $$ TY^0=\{(q,p)\in T^*X\colon q\in Y,p|_{T_qY}=0\}\subseteq T^*X $$ The annihilator of $Y$. Is is it true that the annihilator is a smooth Lagrangian submanifold of $T^*X?$, and how to see this? The given hint is too use a suitable atlas of charts for $X$ but I don't see how this helps me. Do I have to starts which charts on $X$, restrict them to $Y$ and from there on construct charts for $TY^0$?