Let $K$ be a $p$-adic field (a finite extension of the field of $p$-adic numbers ${\mathbb Q}_p$).
Let $T$ be a $K$-torus with character group $X={\sf X}^*(T)$ and cocharacter group $Y={\sf X}_*(T)=X^\vee$.
I would like to have *explicit cocycles* for all Galois cohomology classes in $H^1(K,T)$.

We can compute $H^1(K,T)$ via double duality.

The cup product pairing defines an isomorphism $$ H^1(K,T)\overset\sim\longrightarrow {\rm Hom}\big(H^1(K,X),{\Bbb Q}/{\Bbb Z}\big),$$ where I write $H^1(K,X)$ for $H^1({\rm Gal}(\overline K/K),X)$. See Milne, Arithmetic Duality Theorems, Corollary I.2.3.

Let $L/K$ be a finite Galois extension of degree $d$ splitting $T$. We have $$H^1(K,T)=H^1(\Gamma_{L/K}, Y\otimes_{\Bbb Z} L^\times),$$ where $\Gamma_{L/K}={\rm Gal}(L/K)$. The finite group $\Gamma_{L/K}$ acts on $X$ and $Y$, and we have $$H^1(K,X)=H^1(\Gamma_{L/K}, X).$$ The canonical pairing of $\Gamma_{L/K}$-modules $$ Y\times X\to {\Bbb Z}$$ induces an isomorphism \begin{align*} H^1(\Gamma_{L/K}, X)&\overset\sim\longrightarrow {\rm Hom}\bigg(H^{-2}\big(\Gamma_{L/K},\, Y\otimes_{\Bbb Z} ({\Bbb Q}/{\Bbb Z})\,\big), \ {\Bbb Q}/{\Bbb Z}\bigg)\\ &\overset\sim\longrightarrow {\rm Hom}\big(H^{-1}(\Gamma_{L/K}, Y),{\Bbb Q}/{\Bbb Z}\big); \end{align*} see Brown, Cohomology of Groups, Corollary VI.7.3. Thus we obtain an isomorphism \begin{equation*} \lambda\colon\,(Y_\Gamma)_{\rm tors}= H^{-1}(\Gamma_{L/K}, Y)\overset\sim\longrightarrow H^1(K,T)=H^1(\Gamma_{L/K}, Y\otimes_{\Bbb Z} L^\times), \end{equation*} where $\Gamma={\rm Gal}(\overline K/K)$, $\ Y_\Gamma$ denotes the group of coinvariants of $\Gamma$ in $Y$, and $(\ )_{\rm tors}$ denotes the torsion subgroup of the group in the parentheses.

Question.Let $y\in Y$ be a cocharacter whose image in $Y_\Gamma$ is of finite order (say, of order dividing $d$). Write explicitly a cocycle in $Z^1(\Gamma_{L/K}, Y\otimes_{\Bbb Z} L^\times)$ representing $\lambda[y]\in H^1(\Gamma_{L/K}, Y\otimes_{\Bbb Z} L^\times)$.

I think that this is possible, because I know the corresponding formula for $K={\Bbb R}$.