Local trivialization of the cotangent bundle

Question

In defining the cotangent bundle, Spivak in his differential geometry book, says

Then later,

I don't understand what he means by this last comment. It seems to me that $y_*' \circ (x_*')^{-1}$ and $y_* \circ x_*^{-1}$ don't even have the same domain. How could they be equal?

Answer 1

The key line is in the parentheses:

(We first pick an isomorphism from $(\mathbb{R}^n)^*$ to $\mathbb{R}^n$, once and for all.)

In order for a local trivialization of the cotangent bundle to be a local trivialization, it must map to $U \times \mathbb{R}^n$, so Spivak is implicitly defining his local trivializations via that isomorphism.