I was asked to define explicitly a differentiable structure on $S_1$ such that the inclusion into $\mathbb{R}^2$ is smooth. I defined what I imagine to be the usual differentiable structure on $S_1$ as the atlas generated by the charts $(S_1\setminus\{(0,1)\}, \varphi_1)$ and $(S_1\setminus\{(0,-1)\}, \varphi_2)$ where $\varphi_1$ and $\varphi_2$ are the homeomorphisms between these sets and $\mathbb{R}$ given by trig functions. (I hope these are smooth.)
I'm trying to show that with this structure the inclusion from $S_1$ to $\mathbb{R}^2$ is a $C^\infty$ function. My definition of $C^\infty$ function between manifolds is that a function is $C^\infty$ if the pullback of any smooth function is a smooth function.
The result is very intuitive to me but I can't find a proof.
EDIT:
I've been hinted that better charts and homeomorphisms to use here are:
$$U_1 = \{ (x,y) \in S^1 \mid x > \frac12 \}$$ $$U_2 = \{ (x,y) \in S^1 \mid y > \frac12 \}$$ $$U_3 = \{ (x,y) \in S^1 \mid x < -\frac12 \}$$ $$U_4 = \{ (x,y) \in S^1 \mid y < -\frac12 \}$$
With projections to $x$ for $U_2$ and $U_4$ and projections to $y$ for $U_1$ and $U_3$.