Cohomology of the unit tangent bundle of an even-dimensional sphere
= Cohomology of the unit tangent bundle of an even-dimensional sphere
{title2=$H^*(STS^{2n};\mathbb Z)$}
The Euler class of $TS^{2n}$ evaluates to $2$. Its <Gysin sequence of a sphere bundle> gives
$$
H^q(STS^{2n};\mathbb Z)\cong
\begin{cases}
\mathbb Z,&q=0,4n-1,\\
\mathbb Z/2,&q=2n,\\
0,&\text{otherwise}.
\end{cases}
$$
All products of positive-degree classes vanish.