Manifold parallelizable equivalent condition 
This is an exercise from Loring Tu's Introduction to Manifolds that I am stuck at. I know that a tangent bundle is trivial if it is isomorphic to the product bundle $M \times \mathbb{R}^{n}$. Here $n$ is the dimension of the (smooth of course) manifold $M$.
I think I have to construct a smooth frame from the the isomorphism and vice versa; but I cannot find a way to do so...Could anyone please help me?

If there is a smooth frame $X_1,\dots,X_n$, then each tangent vector $V$ is uniquely written as
$$V=v^1X_1+\dots+v^nX^n.$$ 
What can you say about the map $(x,V)\mapsto(x,v^1,\dots,v^n)$?
