In a post several years ago---
A sine product with (almost) integer values
a user observed that under some certain conditions,
$$ n \left( \prod_{k \in S_{n}} \sin \left( k \frac {\pi}{n} \right) \right)^{-2} $$
is (almost) an integer.
Another user offered a link to an (unnamed) paper by S. Galovich, but the link has since expired. Does anyone know of which of Galovich's papers may have contained theorem(s) pertaining to the above observation? (Galovich is now deceased.) Also, does anyone know of other similar products involving the sine function or any other trigonometric functions that might produce ``almost'' integer values? Thanks.