I was asked to show that $\displaystyle TS^1$ is diffeomorphic to $\displaystyle S^1 \times \mathbb{R}$.

But why can I not just use the map $\displaystyle TS^1 \rightarrow S^1 \times \mathbb{R}$ given by $\displaystyle v \in T_p S^1 \mapsto (p, v)$, since each $\displaystyle T_p S^1 = \mathbb{R}$?

I know this cannot be the correct map, but I am not sure why. Any help would be appreciated.