The tangent bundle $TS^n$ is diffeomorphic the $S^n \times \mathbb{R}^n$? The tangent bundle $TS^n$ is diffeomorphic the $S^n \times \mathbb{R}^n$? 

Comments: I was able to show that $TS^1$ is diffeomorphic the $S^1 \times \mathbb{R}$. For the case $n=2$, I believe it is not true, but I do not know how to justify it.
Hint: It's the hairy ball theorem.
Hint: It's the hairy ball theorem.