TITLE: A sine product with (almost) integer values
QUESTION [5 upvotes]: Let n, k be integers, $n>1$ and $k \perp n$ denote that k, n  are coprime and let $S_n = \{1 \le k \le \lfloor n / 2 \rfloor :  k  \perp n \}.$ Then 
$$ n \left( \prod_{k \in S_{n}} \sin \left( k \frac {\pi}{n} \right) \right)^{-2}  \in \mathbb{Z}. $$
I think this is surprising but I have no proof.

REPLY [3 votes]: There are a number of results for products that are closely connected to your product.  There are some inessential differences (inverse is not taken), and generally products are over all $k$ from $1$ to $n-1$ relatively prime to $n$, but that would be taken care of by your squaring. Here is a link to a fully available paper by Steven Galovich.