I am just posting my comment as one answer. Let $k$ be a field, and let $T$ be a $k$-group scheme such that there exists a field extension $k'/k$ that is finite and separable (i.e., étale), and there exists an isomorphism $i$ of $k'$-group schemes from $T_{k'}:=\text{Spec}\ k'\times_{\text{Spec}\ k}T$ to the split torus $\mathbb{G}_{m,k'}^d$, i.e., $T$ is a torus. By adjointness, there is a natural morphism of $k$-group schemes to the Weil restriction, $T\to R_{k'/k}T_{k'}.$ The composition with $R_{k'/k}i$ gives a morphism of $k$-schemes, $$T\to R_{k'/k}\mathbb{G}_{m,k'}^d.$$ Checking after pullback to $k'$, this is a closed immersion whose quotient, $Q$, is also a torus. Therefore we have a short exact sequence of $k$-group schemes, $$1 \to T \to R_{k'/k}\mathbb{G}_{m,k'}^d \to Q \to 1.$$
The associated long exact sequence of étale cohomology gives the following, $$H^1_{\text{et}}(\text{Spec}\ k',\mathbb{G}_{m,k'})^d \to H^1_{\text{et}}(\text{Spec}\ k,Q) \to H^2_{\text{et}}(\text{Spec}\ k,T)\to H^2_{\text{et}}(\text{Spec}\ k',\mathbb{G}_{m,k'})^d.$$ By Hilbert's Theorem 90, the first group in this sequence is trivial. Standard results about Brauer groups prove that the last group is trivial if and only if the Brauer group of $k'$ is trivial if and only if every $\textbf{PGL}_\ell$-torsor over $k'$ has a $k'$-point (for all positive integers $\ell$).

Thus, the Galois cohomology group $H^2_{\text{et}}(\text{Spec}\ k, T)$ is trivial if every $\textbf{PGL}_\ell$-torsor over $k'$ has a $k'$-point and every $Q$-torsor over $k$-has a $k$-point. For $k$ and $k'$ equal to function fields of curves over an algebraically closed field, this can be proved directly, cf., Tsen's original proof.

In fact, since all such torsors are dense Zariski open subschemes of smooth projective varieties that are separably rationally connected, this also follows from the "rationally connected fibration theorem", i.e., the Kollár-Miyaoka-Mori conjecture.

I am just posting my comment as one answer. Let $K$ be a field, and let $T$ be a $K$-group scheme such that there exists a field extension $K'/K$ that is finite and separable (i.e., étale), and there exists an isomorphism $i$ of $k'$-group schemes from $T_{K'}:=\text{Spec}\ K'\times_{\text{Spec}\ K}T$ to the split torus $\mathbb{G}_{m,K'}^d$, i.e., $T$ is a torus.

By adjointness, there is a natural morphism of $K$-group schemes to the Weil restriction, $T\to R_{K'/K}T_{k'}.$ The composition with $R_{K'/K}i$ gives a morphism of $K$-schemes, $$T\to R_{K'/K}\mathbb{G}_{m,K'}^d.$$ Since it is true after basechange to $K'$, also this morphism of $K$-group schemes is a closed immersion whose quotient, $Q$, is also a torus. Therefore we have a short exact sequence of $K$-group schemes, $$1 \to T \to R_{K'/K}\mathbb{G}_{m,K'}^d \to Q \to 1.$$
The associated long exact sequence of étale cohomology gives the following, $$H^1_{\text{et}}(\text{Spec}\ K',\mathbb{G}_{m,K'})^d \to H^1_{\text{et}}(\text{Spec}\ K,Q) \to H^2_{\text{et}}(\text{Spec}\ K,T)\to H^2_{\text{et}}(\text{Spec}\ K',\mathbb{G}_{m,K'})^d.$$ By Hilbert's Theorem 90, the first group in this sequence is trivial. Standard results about Brauer groups prove that the last group is trivial if and only if the Brauer group of $K'$ is trivial if and only if every $\textbf{PGL}_\ell$-torsor over $K'$ has a $K'$-point (for all positive integers $\ell$).

Thus, the Galois cohomology group $H^2_{\text{et}}(\text{Spec}\ K, T)$ is trivial if every $\textbf{PGL}_\ell$-torsor over $K'$ has a $K'$-point and every $Q$-torsor over $K$-has a $K$-point. For $K$ and $K'$ equal to function fields of curves over an algebraically closed field $k$, this can be proved directly, cf., Tsen's original proof.

In fact, since all such torsors are dense Zariski open subschemes of smooth projective varieties that are separably rationally connected, this also follows from the "rationally connected fibration theorem", i.e., the Kollár-Miyaoka-Mori conjecture.

I am just posting my comment as one answer. Let $k$ be a field, and let $T$ be a $k$-group scheme such that there exists a finite, Galois field extension $k'/k$ and an isomorphism $i$ of $k'$-group schemes from $T_{k'}:=\text{Spec}\ k'\times_{\text{Spec}\ k}T$ to the split torus $\mathbb{G}_{m,k'}^d$, i.e., $T$ is a torus. By adjointness, there is a natural morphism of $k$-group schemes to the Weil restriction, $T\to R_{k'/k}T_{k'}.$ The composition with $R_{k'/k}i$ gives a morphism of $k$-schemes, $$T\to R_{k'/k}\mathbb{G}_{m,k'}^d.$$ Checking after pullback to $k'$, this is a closed immersion whose quotient, $Q$, is also a torus. Therefore we have a short exact sequence of $k$-group schemes, $$1 \to T \to R_{k'/k}\mathbb{G}_{m,k'}^d \to Q \to 1.$$ Let $k'\hookrightarrow \overline{k}$ denote a separable closure. Denote by $\Gamma$ the Galois group $\text{Gal}(\overline{k}/k)$, and denote by $\Gamma'\lhd \Gamma$ the Galois group $\text{Gal}(\overline{k}/k')$.
The associated long exact sequence of Galois cohomology gives the following, $$\dots \to H^1_{\text{pro}}(\Gamma',\overline{k}^\times)^d \to H^1_{\text{pro}}(\Gamma,Q(\overline{k})) \to H^2_{\text{pro}}(\Gamma,T(\overline{k}))\to H^2(\Gamma',\overline{k}^\times)^d\to \dots$$ By Hilbert's Theorem 90, the first Galois cohomology group vanishes. Standard results about Brauer groups prove that the last Galois cohomology group vanishes if and only if the Brauer group of $k'$ is trivial if and only if every $\textbf{PGL}_\ell$-torsor over $k'$ has a $k'$-point (for all positive integers $\ell$.

Thus, the Galois cohomology group $H^2(\Gamma, T(\overline{k}))$ is trivial if every $\textbf{PGL}_\ell$-torsor over $k'$ has a $k$'-point and every $Q$-torsor over $k$-has a $k$-point. For $k$ and $k'$ equal to function fields of curves over an algebraically closed field, this can be proved directly, cf., Tsen's original proof. In fact, since all such torsors are dense Zariski open subschemes of smooth projective varieties that are separably rationally connected, this also follows from the "rationally connected fibration theorem", i.e., the Kollár-Miyaoka-Mori conjecture.

I am just posting my comment as one answer. Let $k$ be a field, and let $T$ be a $k$-group scheme such that there exists a field extension $k'/k$ that is finite and separable (i.e., étale), and there exists an isomorphism $i$ of $k'$-group schemes from $T_{k'}:=\text{Spec}\ k'\times_{\text{Spec}\ k}T$ to the split torus $\mathbb{G}_{m,k'}^d$, i.e., $T$ is a torus. By adjointness, there is a natural morphism of $k$-group schemes to the Weil restriction, $T\to R_{k'/k}T_{k'}.$ The composition with $R_{k'/k}i$ gives a morphism of $k$-schemes, $$T\to R_{k'/k}\mathbb{G}_{m,k'}^d.$$ Checking after pullback to $k'$, this is a closed immersion whose quotient, $Q$, is also a torus. Therefore we have a short exact sequence of $k$-group schemes, $$1 \to T \to R_{k'/k}\mathbb{G}_{m,k'}^d \to Q \to 1.$$
The associated long exact sequence of étale cohomology gives the following, $$H^1_{\text{et}}(\text{Spec}\ k',\mathbb{G}_{m,k'})^d \to H^1_{\text{et}}(\text{Spec}\ k,Q) \to H^2_{\text{et}}(\text{Spec}\ k,T)\to H^2_{\text{et}}(\text{Spec}\ k',\mathbb{G}_{m,k'})^d.$$ By Hilbert's Theorem 90, the first group in this sequence is trivial. Standard results about Brauer groups prove that the last group is trivial if and only if the Brauer group of $k'$ is trivial if and only if every $\textbf{PGL}_\ell$-torsor over $k'$ has a $k'$-point (for all positive integers $\ell$).

Thus, the Galois cohomology group $H^2_{\text{et}}(\text{Spec}\ k, T)$ is trivial if every $\textbf{PGL}_\ell$-torsor over $k'$ has a $k'$-point and every $Q$-torsor over $k$-has a $k$-point. For $k$ and $k'$ equal to function fields of curves over an algebraically closed field, this can be proved directly, cf., Tsen's original proof.

In fact, since all such torsors are dense Zariski open subschemes of smooth projective varieties that are separably rationally connected, this also follows from the "rationally connected fibration theorem", i.e., the Kollár-Miyaoka-Mori conjecture.