Using complex numbers prove that $\sin(\frac {\pi}{m})\sin(\frac {2\pi}{m})\ldots\sin(\frac {(m-1)\pi}{m})= \frac{m}{2^{m -1}}$

Question

Prove that for $m=2,3,\dots$ $$ \sin\left(\frac{\vphantom{1}\pi}m\right)\sin\left(\frac{2\pi}m\right)\sin\left(\frac{3\pi}m\right)\cdots\sin\left(\frac{(m-1)\pi}m\right)=\frac{m}{2^{m-1}} $$

I have no idea how to begin at all, I'm trying to think of a way using de Moivre's theorem but I can't seem to figure it out.
I'm sorry for not showing effort but I'm completely stuck.

Answer 1

Generalized from this answer: $$ \begin{align} \prod_{k=1}^{m-1}\left(\frac{e^{i\pi\frac km}-e^{-i\pi\frac km}}{2i}\right) &=\frac{e^{i\pi\frac{m(m-1)}{2m}}}{2^{m-1}i^{m-1}}\prod_{k=1}^{m-1}\left(1-e^{-i2\pi\frac km}\right)\\ &=\frac{e^{i\pi\frac{m-1}2}}{2^{m-1}i^{m-1}}\lim_{z\to1}\prod_{k=1}^{m-1}\left(z-e^{-i2\pi\frac km}\right)\\[3pt] &=\frac1{2^{m-1}}\lim_{z\to1}\frac{z^m-1}{z-1}\\[9pt] &=\frac{m}{2^{m-1}} \end{align} $$ The statement in the question seems to be missing a pair of braces: \frac{m}{2^m-1} rather than \frac{m}{2^{m-1}}.

Answer 2

In this case left hand side of equation is not equal to right hand sides of equation .You can try it by putting random values of m.

Example ($m=3$): $$\sin \frac{\pi}{3} \sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2}\frac{\sqrt{3}}{2} = \frac{3}{4} \neq \frac{3}{2^3 - 1}$$

Answer 3

$|\cos \theta + i\sin \theta - 1|\\ \sqrt {1 - 2\cos \theta} = 2\sin \frac {\theta}{2}\\ 2\sin\theta = |e^{2\theta i} - 1|$

$\prod_\limits{n=1}^{m-1} \sin \frac{n\pi}{m} = \frac {1}{2^{m-1}}\prod_\limits{n=1}^{m-1}|e^{\frac{2n\pi}{m} i} - 1|$

The set $\{e^{\frac{2n\pi}{m} i}\}$ make up the roots of $z^m - 1 = 0$ excluding the root at $z= 1$ or the roots of $(z^{m-1} + z^{m-2} + z^{m-3} + \cdots + 1)$

$(z - e^{\frac{2\pi}{m}i})(z - e^{\frac{4\pi}{m}i})(z - e^{\frac{6\pi}{m}i})\cdots(z - e^{\frac{(m-2)\pi}{m}i})= (z^{m-1} + z^{m-2} + z^{m-3} + \cdots + 1)\\ |z - e^{\frac{2\pi}{m}i}||z - e^{\frac{4\pi}{m}i}||z - e^{\frac{6\pi}{m}i}|\cdots|z - e^{\frac{(m-2)\pi}{m}i}|= |z^{m-1} + z^{m-2} + z^{m-3} + \cdots + 1| $

and now set z = 1

$|1 - e^{\frac{2\pi}{m}i}||1 - e^{\frac{4\pi}{m}i}||1 - e^{\frac{6\pi}{m}i}|\cdots|1 - e^{\frac{(m-2)\pi}{m}i}| = |1^{m-1} + 1^{m-2} + 1^{m-3} + \cdots + 1| = m$

$\prod_\limits{n=1}^{m-1} \sin \frac{n\pi}{m} = \frac {1}{2^{m-1}}\prod_\limits{n=1}^{m-1}|e^{\frac{2n\pi}{m} i} - 1| = \frac {m}{2^{m-1}}$