Let $K$ be a number field and $K_v$ be the completion of $K$ at the place $v$. Consider an elliptic curve $E/K$ over $K$. The short exact sequence $0 \to E[2] \to E \to E \to 0$ induces a famous exact sequence $0 \to E(K)/2E(K) \to H^1(G_K, E[2]) \stackrel{\delta}{\to} H^1(G_{K_v}, E)[2] \to 0$.
For $M$: discrete p-primary abelian group $M$, we let $\widehat{M}=\text{Hom}(M, \Bbb{Q}_p/\Bbb{Z}_p)$ be the Pontryagin dual of $M$.
My question is, why is the image of $\delta$ isomorphic to $\widehat{H^1(G_{K_v}, E)[2]}$?
This result can be deduced from a stronger result in 'Galois Cohomology of Elliptic Curves' by J. Coates and R. Sujatha, but I am interested in a direct proof or approach to this question.