# Tangent bundle of a product of smooth manifolds diffeomorphic to product of the tangent bundles of manifolds

How can I prove that the tangent bundle of a product of smooth manifolds are diffeomorphic to the product of the tangent bundles of manifolds? Further, how can I deduce it to the fact that a tangent bundle of a torus $\mathbb S^1 \times \mathbb S^1$ is diffeomorphic to $\mathbb S^1 \times \mathbb S^1 \times \mathbb R^2$?

Some hint or approach would be much appreciated, I am stuck right at the beginning.

• There's a canonical map between the two. All you have to do is show it's a diffeomorphism. – Matthew Leingang Nov 15 '17 at 12:55