Let $S^{2n+1}$ be the unit sphere considered as a subset of $\mathbb{C}^{n+1}$ and define an action of $S^1 \subset \mathbb{C}$ by usual (complex) scalar multiplication. The quotient space is the well known complex projective space $\mathbb{C}P^n$. Denote by $\pi : S^{2n+1} \to \mathbb{C}P^n$ the natural projection and by $[z_1 : \cdots : z_{n+1}]$ the image of $(z_1, \dots, z_{n+1})$ under $\pi$.
For $1 \leq i \leq n+1$, let $U_i = \{ [z_1 : \cdots : z_{n+1}] \in \mathbb{C}P^n : z_i \neq 0\}$ and endow $\mathbb{C}P^n$ with a differentiable structure making the $U_i$ open submanifolds.
I already proved $\pi^{-1}(U_i)$ is homeomorphic to $U_i \times S^1$ via the map $\psi_i : \pi^{-1}(U_i) \to U_i \times S^1$ given by $\psi_i(z) = \left( \pi(z), \frac{z_i}{|z_i|} \right)$.
I am now trying to prove $\psi_i^{-1} : U_i \times S^1 \to \pi^{-1}(U_i)$ is differentiable. A formula for $\psi_i^{-1}$ is known:
$\psi_i^{-1}(x, a) = \frac{a |z_i|}{z_i} \cdot z$, for a chosen $z \in \pi^{-1}(x)$.
Am I overthinking this or it is obvious?