From general topology, I know that the two mentioned spaces are homeomorphic: one can see $\mathbb{RP}^1$ as $S^1$ equiped with the antipodal quotient relation. Then one finds a (surjective) map from $S^1$ to itself which respects this relation, such as $z\in S^1\subset\mathbb{C}\mapsto z^2\in S^1\subset\mathbb{C}$ and conludes by the universal property of the quotien topology (plus the fact $S^1$ is compact Hausdorff) that this mapping induces an homeomorphism $h:S^1/_\sim\cong\mathbb{RP}^1\rightarrow S^1$.
The fact is, to prove that $h$ is a diffeomorphism, as we have not studied something as quotient differentiable structures (by the way, does something like this even exist?) I would need the explixit definition of $h$. I thought of $[x_0:x_1]\in\mathbb{RP}^1\mapsto \frac{(x_0+ix_1)^2}{x_0^2+x_1^2}\in S^1$ but this is not even well defined.
Any help would be great