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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.

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

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