I am trying to show that it is not possible, unless $\theta$ is an integer, that $\sin n\pi\theta$ or $\cos n\pi\theta$ should be nearly equal, for all large $n$, to one or other of two values $a$, $b$.
I am unsure if the wording of the question is fully clear to me. Can the hypothesis be stated as $\cos n\pi\theta - \sin n\pi\theta \rightarrow 0$ only when $\theta \in Z$.
Any help or clues would be highly appreciated.