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([itex]TS^1[/itex] is the tangent bundle of the 1-sphere.)

We can use the theorem stating the following.

*If [itex]M[/itex] is a smooth [itex]n[/itex]-manifold with or without boundary, and [itex]M[/itex] can be covered by a single smooth chart, then [itex]TM[/itex] is diffeomorphic to [itex]M×ℝ^n.[/itex]*

Clearly, I must be looking for a single smooth chart on [itex]S^1[/itex], but I am very uncertain on how to go about doing this. Any tips on managing the nuances are greatly appreciated.