TITLE: Trivial Tangent and Cotangent Bundles
QUESTION [5 upvotes]: If we have a smooth manifold $M$, why is the tangent bundle $TM$ trivial (as a vector bundle) iff the cotangent bundle $T^*M$ is trivial as well?

REPLY [6 votes]: Hint: Given a global frame for either, there is a pretty natural way to create a global frame for the other by dualizing or taking the "covector".