Proving that $(T S^n) \times \mathbb R$ is diffeomorphic to $S^n\times \mathbb R^{n+1}$

Question

I'm asked to show a diffeomorphism $\psi$ between $(T S^n) \times \mathbb R$ and $S^n\times \mathbb R^{n+1}$.

I'm not sure on how to proceed here. I have written down $(T S^n) \times \mathbb R$ and $S^n\times \mathbb R^{n+1}$ as

$$(T S^n) \times \mathbb R= \big\{ (x, v, r): x\in S^n, v\in x^\perp, r\in \mathbb R \big\}, $$ $$S^n\times \mathbb R^{n+1}= \big\{ (x, w): x\in S^n, w\in \mathbb R^{n+1} \big\} $$

Can you help me in finding such diffeomorphism?