Product of sines

Question

I am looking to evaluate

$$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n}$$

without using complex numbers. I can show the result if $n$ is a power of $2$, but if $n$ is anything else I reach a point where I can no longer proceed.

My current method (for when $n$ is a power of $2$) is to note that all of the angle that sin is being evaluated on are evenly distributed between $0$ and $\pi$, but more importantly, one of the angles will always be $\frac{\pi}{2}$ and all angles which are greater than $\frac{\pi}{2}$ can be re-written as the cos of an angle that is less than $\frac{\pi}{2}$ (using the complementary angle). When this is done, we get

$$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \prod_{k = 1}^{\lfloor\frac{n - 1}{2}\rfloor}\frac{1}{2}\sin\frac{2k\pi}{n} = \frac{1}{2^{\lfloor\frac{n - 1}{2}\rfloor}}\prod_{k = 1}^{\lfloor\frac{n - 1}{2}\rfloor}\sin\frac{k\pi}{\frac{n}{2}}$$

This process can then be repeated recursively until we have just one $\sin$ term left in the product (which will be $\sin\frac{\pi}{2}$), then what is left is equivalent to

$$\frac{n}{2^{n - 1}}\;.$$

Can anyone suggest a way of generalizing this to when $n$ is not a power of $2$?

Thanks in advance.

Answer 1

For any $z\in(0,1)$ we have $\sin(\pi z)=\frac{\pi}{\Gamma(z)\,\Gamma(1-z)}$, hence your question boils down to:

Q.: How can we prove the multiplication formula for the $\Gamma$ function using real-analytic tools?

The answer is given on PlanethMath: we may prove it through the functional relation $\Gamma(z+1)=z\,\Gamma(z)$ and Stirling's approximation. For an elementary proof of Stirling's approximation, have a look at this answer of mine: it can be done through creative telescoping and the above reflection formula.