I've been having a hard time trying to determine if the tangent bundle of a differentiable manifold is trivial. Namely, if there exists a diffeomorphism between the tangent bundle $TM$ of a given manifold $M$ and the product manifold of $M\times \mathbb{R}^n$.
I've managed to build a diffeomorphism from $TS^1$ to $S^1\times \mathbb{R}^1$. But the case with torus $S^1\times S^1$ seems harder, since the dimension is higher.
In general, how do I show that $S^1\times \cdots \times S^1$ has trivial tangent bundle?