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I'm stuck with this problem:

"Let us consider, on the sphere, the local coordinates associated with the differential structure determined by the atlas $\{(U_1=S^n\setminus\{N\},\varphi_N),(U_2=S^n\setminus\{S\},\varphi_S)\}$ where $\varphi_N,\varphi_S$ are the two stereographic projections from the north pole and the south pole, respectively. Calculate the representation in these coordinates of the map $p_n:S^1\rightarrow S^1$, $z\mapsto z^n$, with $n\in\mathbb{Z}$ and prove that $p_n$ is smooth"

I first tryied with $z=(x,y)$ such that $x^2+y^2=1$, but the map was not differentiable, so my teacher suggest me to try with the angle function, but I get stuck in it.

The angle function is defined like $\theta :U\subset S^1\rightarrow \mathbb{R}$ such that for all $p\in U$ $e^{i\theta(p)}=p$.

What can I do to solve it?

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