Cup product of inhomogeneous Tate cochains, and application to tori over local fields that split over cyclic extensions

In this note we give formulas for cup product in Tate cohomology in terms of inhomogeneous cochains. Using one of these formulas, for a torus T defined over a non-archimedean local field K and splitting over a cyclic extension of K, we compute explicit cocycles representing all cohomology classes in H^1(K,T).