A version of local Tate duality stated: Let $K$ be a finite extension of $\mathbb Q_p$, $A$ be a finite $G_K=Gal(\overline K/K)$ module. Then for $0\le i\le 2$, the cup product induces a perfect pairing $H^i(G_K,A)\times H^{2-i}(G_K,A')\to \mathbb Q/\mathbb Z$, where $A'=Hom(A,\overline K^\times)$. In particular, there is a canonical isomorphism $$H^i(G_K,A)\cong H^{2-i}(G_K,A')^\vee,$$ where $^\vee=Hom(\cdot,\mathbb Q/\mathbb Z).$

However, another version is used when studying the global/local deformation ring: Let $K$ be a finite extension of $\mathbb Q_p$, $\mathbb F$ be its residue field, and $V$ be a finite dimensional $\mathbb F$-vector space with a continuous $G_K$ action. Let $V^*$ be the dual representation. Then for $0\le i\le 2$, there is a canonical isomorphism $$H^i(G_K,V) \cong H^{2-i}(G_K,V^*(1))^*. $$

Can I ask whether it is possible to derive the second version from the first one? If not, is there a reference for the proof of the second version?

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