Q13/025) Prove that sin(1degree) is not a rational number?

we have

cos 2 t = 1 - 2 sin ^2 t

so if sin 1 is rational cos 2 is rational

now cos 0 =1 is rational

cos (n-2 ) + cos (n+2) = 2 cos n cos 2

so cos (n+2) = - cos (n-2) + 2 cos n cos 2

if cos n and cos n- 2 are rational then by strong induction cos n+2 is rational

hence proceeding we get cos 30 = sqrt(3)/2 is rational which is contradiction

hence cos 2 and then sin 1 are not rational