Let $k_n=\mathbb{Q}_p(\zeta_{p^n})$ and $T$ be the $p$-adic Tate module of an elliptic curve $E$. Assume $E$ has good supersingular reduction at $p$. Let $V=T\otimes \mathbb{Q}_p$. Now, by Local Tate Duality, there is a perfect pairing,
$$H^1(k_n, V/T)\times H^1(k_n, T)\longrightarrow \mathbb{Q}_p/\mathbb{Z}_p$$
See the following image:
Here, it is mentioned that to show that the corestriction maps are surjective it suffices to show that the restriction maps are injective by Tate Duality. I do not understand how injectivity$\implies$surjectivity in this case. Can someone help me understand this please!