This most likely follows from a standard result but a lack of knowledge prevents me from seeing this.
Let $K$ be a non Archimedean local field. Let $\Gamma$ be $\mathrm{Gal}(\bar{k}/k)$. Let $T$ be a torus over $k$. Then define $\mathrm{X}(T)= \mathrm{Hom}(T,\mathbb{G}_{m})$ and this has a continuous $\Gamma$ action. Further let $\mathrm{X}(T)_{\mathbb{Q}} = \mathrm{X}(T) \otimes _{\mathbb{Z}}\mathbb{Q}$.
Then why is $\mathrm{H^{2}}(\Gamma, \mathrm{X}(T)_{\mathbb{Q}})$ trivial. Is it because $\mathrm{X}(T)_{\mathbb{Q}}$ is an injective $\Gamma$ module? Any reference will be helpful.