Let $E$ be an elliptic curve over $\mathbb{Q}_p$ and $k_n=\mathbb{Q}_p({\zeta_{p^n}})$. Let $T$ be the $p$-adic Tate module of $E$. Then there is a local Tate pairing, $$<,>_n: H^1(k_n, E[p^\infty])\times H^1(k_n,T)\longrightarrow \mathbb{Q}_p/\mathbb{Z}_p$$
I can't seem to figure out how this pairing is defined, and how the map can be induced from the Weil pairing. I have tried to deduce from Tate duality, but I don't understand why there is $E[p^\infty]$ in the pairing.
If someone can please help and provide a hint, it'll be greatly appreciated. Thank you!