Professor Lee's book on Smooth Manifolds has the following problem:
Show that $p_n: \mathbb{S}^1 \to \mathbb{S}^1$ for $n\in \mathbb{Z}$, given in complex notation by $p_n(z)=z^n$, is a smooth map.
I have a proof which uses angle charts. I have another idea for a proof and I'm unsure if such a solution is correct. Please check for mistakes.
Proof:
Let $M$ be a smooth manifold and $f,g$ smooth maps $M\xrightarrow{f,\\ g} \mathbb{S}^1$. If we show that the standard complex multiplication $\mathbb{S}^1 \times \mathbb{S}^1 \to \mathbb{S}^1$ is smooth, then so is the product $f\cdot g$ which factors as $$M\xrightarrow{f \times g} \mathbb{S}^1 \times \mathbb{S}^1 \xrightarrow{\cdot} \mathbb{S}^1,$$ is smooth by composition. $f\times g$ is smooth since each component function is. Hence to show that $p_n$ is smooth, it is enough to show that $p_2$ is smooth.