2015/036) Prove that $\sin(1^\circ)$ is not a rational number

we have

$\cos 2 t= 1 - 2 sin ^2 t$

so if $\sin\, 1^\circ$ is rational $\cos\, 2\circ$ is rational

now $\cos\ 0^\circ =1$ is rational

$\cos (n-2^\circ ) + \cos (n+2^\circ) = 2 \cos\, n \cos 2^\circ$

so $\cos (n+2^\circ) = - \cos (n-2^\circ) + 2 \cos\, n \cos\, 2^\circ$

if $cos\, n$ and $cos (n- 2^\circ)$ are rational then by strong induction $cos (n+2^\circ)$ is rational

hence proceeding we get $cos\,30^\circ = \dfrac{\sqrt3}{2}$ is rational which is contradiction

hence $\cos\,2^\circ$ and then $\sin\,1^\circ$ are not rational