Trivializing the vector bundle of $S^n$

Question

It's know that in general we do not have that $TS^n\cong S^n \times \mathbb{R}^n$. But I saw the interesting fact that is that we will have $TS^n \times \mathbb{R}\cong S^n \times \mathbb{R}^{n+1}$, so we get kinda of a trivialization . Now I have been trying to see why this is true but I got nowhere, does anyone have any suggestions ? Thanks in advance.

Answer 1

Consider the standard embedding $\iota:S^n\to\mathbb{R}^{n+1}$, and in particular the tangent/normal splitting $T\mathbb{R}^{n+1}|_{S^n}\cong TS^n\oplus NS^n$. Can you use this to show your statement is true?