We have $S^2+C^2=\sin^2{64x}+\sin^2{65x}+\cos^2{64x}+\cos^2{65x}+2(\sin{64x}\sin{65x}+\cos{64x}\cos{65x})
=2+2\cos{x}$, thus $\cos{x}$ is rational. Now repeated use of the formula $\cos{2y}=2\cos^2{y}-1$ on $\cos{64x}$, gives $\cos{64x}$ as rational, and since $C$ is rational $\cos{65x}$ will also be rational.
Notice that $$S^2+C^2=2=2\sin 64x \sin 65x+2\cos 64x \cos 65x=2+2\cos (65-64)x,$$so $\cos x$ is rational. Evidently, $\cos 2^nx=2\cos^2 2^{n-1}x-1$ so $\cos 64x$ is rational and so $\cos 65x$ is rational as well since $C \in \mathbb{Q}$.