Poking around, I find several questions and answers about products of sines where the indexed coefficient is in the numerator (e.g. $\prod_i \sin{\frac{a_i\pi}{n}}$) or is an integer in the denominator (e.g. $\prod_i \sin{\frac{n\pi}{i}}$).
I am looking for a closed form expression for a product of sines where the indexed coefficient is a prime number in the denominator, e.g. $\prod_i \sin{\frac{n\pi}{p_i}}$.
I'm not asking anyone to derive an expression, just to point me to an answer if one is already known.
Added by edit: In response to query in comments, I am interested in the product running to $p_i< n$. Note that if $n$ is a composite integer, or if it is a prime integer and $p_i$ runs up to $n$, then one of the arguments will be an integer times $\pi$, the $\sin$ of which will be $0$ and hence the entire product will be $0$. In order to obtain a closed form, one should consider $n$ to be a real number, and then see if the closed form behaves as expected when $n$ is set to a composite integer.