Dihedral groups non-commutavity

Question

Here is a result about dihedral groups. $rs = sr ^{-1}$, where $r$ is a rotation of $\frac{2 \pi}{n}$ radians and $s$ is a reflection about the line of symmetry from vertex $i$ and the origin. This is quite understable, and I can basically prove this using my intuition, but I cannot reach a rigorous proof. I also thought that as this result is familiar, so I searched all the questions in MSE but I did not find any. So I posted this question. Can you please help?

Answer 1

Any reflection is its own inverse. Since $rs$ is a reflection we have $(rs)^{-1}=rs$. On the other hand we also know that $(rs)^{-1}=s^{-1}r^{-1}=sr^{-1}$. ($s=s^{-1}$ because $s$ is a reflection as well). Since the inverse is unique we conclude that $rs=sr^{-1}$.