I need to show that there is such a diffeomorphism between these spaces. I've tried looking at the 'faces' of elements on both spaces. It went like this: every element in $T\mathbb{S}^n\times \mathbb{R}$ is of the form $([x,i,\vec{v}],r)$, where $x\in\mathbb{S}^n$, $i\in\Gamma$ is an index from the atlas $\Phi$ of $x\in\mathbb{S}^n$, $\vec{v}\in\mathbb{R}^n$ and $r$ is real.
Elements in $\mathbb{S}^n\times\mathbb{R}^{n+1}$ are of the form $(x,\vec{w})$, $x\in\mathbb{S}^n$ and $\vec{w}\in\mathbb{R}^{n+1}$. This is more conveniently written as $\mathbb{S}^n\times\mathbb{R}^{n+1}=\mathbb{S}^n\times\mathbb{R}^{n}\times\mathbb{R}$ so its elements are like $(x,\vec{v},r)$, $\vec{v}\in\mathbb{R}^n$.
Now, I just take the projection $$ \begin{array}{llll} p:&T\mathbb{S}^n\times\mathbb{R}&\longrightarrow&\mathbb{S}^n\times\mathbb{R}^n\times\mathbb{R}\\ &([x,i,\vec{v}],r)&\longmapsto& (x,\vec{v},r) \end{array} $$
It's not hard to prove that this projection is well defined (since it's defines on the equivalence class $[x,i,\vec{v}]$.
But now, how can I proof that this is a diffeomorphism? It seems obvious that is is invertible but I'm unsure how to proceed on differentiability! My intention is to use the inverse function theorem (since I already have injectivity).
Or is there any easier, more direct way to do this?
Both of them are naturally diffeomorphic to $T\Bbb{R}^{n+1}|_{S^n}$.