I am looking at page 2 of the lecture notes
http://staff.ustc.edu.cn/~wangzuoq/Courses/16F-Manifolds/Notes/Lec15.pdf.
Let $\{U_j\}_{1\leq j \leq n+1}$ denote the usual open covering of $\mathbb{CP}^{n}$ and let $\pi: S^{2n+1} \to \mathbb{CP}^n$ denote the natural projection. The author constructs local trivializations by defining $$ \Phi_{j}: \pi^{-1}\left(U_{j}\right) \rightarrow U_{j} \times S^{1}, \quad \Phi_{j}(z)=\left(\left[z^{1}: \cdots: z^{n+1}\right], \frac{z^{j}}{\left|z^{j}\right|}\right). $$
The author then claims that the inverse of this map is given by $$ \Phi_{j}^{-1}\left(\left[z^{1}: \cdots: z^{n+1}\right], e^{i \theta}\right)=\frac{e^{i \theta}\left|z^{j}\right|}{\sqrt{\left|z^{1}\right|^{2}+\cdots+\left|z^{n+1}\right|^{2}} z^{j}}\left(z^{1}, \cdots, z^{n+1}\right). $$ Is this well-defined, i.e., invariant under scaling by elements of $S^1$? It doesn't look so to me, but maybe I'm missing something.
Do you mean this?
$\Phi_{j}^{-1}\left(\left[z \cdot z^{1}: \cdots: z \cdot z^{n+1}\right], e^{i \theta}\right)=\frac{e^{i \theta}\left|z \cdot z^{j}\right|}{\sqrt{\left|z \cdot z^{1}\right|^{2}+\cdots+\left|z \cdot z^{n+1}\right|^{2}} z \cdot z^{j}}\left(z \cdot z^{1}, \cdots, z \cdot z^{n+1}\right) = \frac{e^{i \theta}\left|z^{j}\right|z^2}{z^2\sqrt{\left|z^{1}\right|^{2}+\cdots+\left|z^{n+1}\right|^{2}} z^{j}}\left(z^{1}, \cdots, z^{n+1}\right) = \frac{e^{i \theta}\left|z^{j}\right|}{\sqrt{\left|z^{1}\right|^{2}+\cdots+\left|z^{n+1}\right|^{2}} z^{j}}\left(z^{1}, \cdots, z^{n+1}\right) = \Phi_{j}^{-1}\left(\left[z^{1}: \cdots: z^{n+1}\right], e^{i \theta}\right)$