How deduce $\prod_{k=1}^{(n-1)/2} \sin^2 \frac{k\pi}{n} =\frac{n}{2^{n-1}}$ from $\prod_{k=1}^{n-1} \sin \frac{k\pi}{n} =\frac{n}{2^{n-1}}$?

Question

I know that $\displaystyle\prod_{k=1}^{n-1} \sin \frac{k\pi}{n} =\frac{n}{2^{n-1}}$ for any integer $n \geq 1$ is true.

Now, suppose that $n$ is odd, how show $$ \prod_{k=1}^{(n-1)/2} \sin^2 \frac{k\pi}{n} =\frac{n}{2^{n-1}} ? $$ I read in a book (Bourbaki) that the first equality implies the second.