Let $ \cos 1^{\circ}$ be rational. Then so is $ \cos 2^{\circ}$. Again we have $ \cos (n+1)^{\circ}+\cos (n-1)^{\circ}=2\cos n^{\circ}\cos 1^{\circ}$.
So using strong induction we get that $ \cos n^{\circ}$ is rational for all integers $ n\geq1$.
This is absurd as $ \cos 30^{\circ}$ is not rational.
Hence our assumption is not correct. So the proof follows.