The Brauer group of tori

We show that the map $\mathrm{Br}T\to (\mathrm{Br}{T}_{\bar{k}}{)}^{{\mathrm{\Gamma }}_k}$ is surjective for a torus $T$ defined over a field $k$ of characteristic $0$ when $k$ is a local or global field or $T$ is quasi-trivial.