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.