How to prove sine product formula?

Question

I am trying to prove this formula:

$\prod_{k=1}^{n-1} \sin (\frac{\pi k}{2n}) = \frac{\sqrt{n}}{2^{n-1}}$

I've tried few approaches:

1. Taking log on it and transforming product to sum of logs.

2. Coupling $\sin \frac{\pi k}{2n} \sin \frac{\pi (n-k)}{2n}=\frac{1}{2} \sin \frac{\pi k}{n}$

3. Proof by induction (worked, but i need proof with complex numbers)

4. Some weird approaches

But nothing worked for me(except induction). I know about sine product formula, but we are just learning complex numbers, i don't want to go that hard. I need to prove it using complex numbers, like taking Im from this expression or something else.