# Untrodden path in “An exotic sequence”?

The post An exotic sequence was so popular that @Noam D. Elkies responded. There's one part of it that I don't understand and it's a question in its own right.

It was enough back there to prove that the sequence $n\tan^{-1} \sqrt 7$ is arbitrarily close to $m\pi$, for integers $m$.

1) Would it suffice to show that $\dfrac{\pi}{\tan^{-1} \sqrt 7}$ is irrational? I suspect so, since for any irrational $\alpha$, the famous result is that sequence $n\alpha$ is dense mod 1.

2) What difficulties are there in proving that $\dfrac{\pi}{\tan^{-1} \sqrt 7}$ is irrational? Now I'm unable to do it myself, but are there possible paths that can be comfortably taken, in a proof by hand?