How to prove those "curious identities"?

Question

How to prove $$ \prod_{k=1}^{n-1} \sin\left(\frac{k\pi}{n}\right) = \frac{n}{2^{n-1}}$$ and $$ \prod_{k=1}^{n-1} \cos\left(\frac{k\pi}{n}\right) = \frac{\sin(\pi n/2)}{2^{n-1}}$$

Answer 1

For the first: $$ \lim_{z=1}\frac{z^n-1}{z-1}=n\tag{1a} $$ $$ \frac{z^n-1}{z-1}=\prod_{k=1}^{n-1}(z-e^{2\pi ik/n})\tag{1b} $$ $$ |1-e^{i2k\pi/n}|=|2\sin(k\pi/n)|\tag{1c} $$ Combining $(1a)$, $(1b)$, and $(1c)$, we get $$ 2^{n-1}\prod_{k=1}^{n-1}\sin(k\pi/n)=n $$ since everything is positive.

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For the second:

If $n$ is even, then $\cos(\frac{\pi}{2})=0$ appears in the product (when $k=n/2$) and $\sin(\frac{n\pi}{2})=0$.

If $n$ is odd, then combining $$ \lim_{z=1}\frac{z^n+1}{z+1}=1\tag{2a} $$ $$ \frac{z^n+1}{z+1}=\prod_{k=1}^{n-1}(z+e^{2\pi ik/n})\tag{2b} $$ $$ 1+e^{i2k\pi/n}=2\cos(k\pi/n)e^{ik\pi/n}\tag{2c} $$ and noting that $\displaystyle\sum_{k=1}^{n-1}k=\frac{n(n-1)}{2}$ so that $\displaystyle\prod_{k=1}^{n-1}e^{ik\pi/n}=(-1)^{(n-1)/2}$ which matches the sign of $\sin(\pi n/2)$, yields $$ 2^{n-1}\prod_{k=1}^{n-1}\cos(k\pi/n)=(-1)^{(n-1)/2}=\sin(\pi n/2) $$

Answer 2

Denote $w = e^{i \pi/n}$. We have

$$\prod_{k = 1}^{n-1} \sin \left(\frac{k\pi}{n}\right)= \prod_{k = 1}^{n-1} \frac{w^k - w^{-k}}{2i} = \frac{1}{2^{n-1}} \prod_{k = 1}^{n-1} \frac{w^k}{i} (1-w^{-2k})$$

Since we have

$$\sum_{k = 0}^{n-1} x^k = \prod_{k = 1}^{n-1} (x-w^{2k})$$

Setting $x=1$ yields

$$\prod_{k = 1}^{n-1} (1-w^{2k}) = n$$

So we get

$$\prod_{k = 1}^{n-1} \sin \left(\frac{k\pi}{n}\right)= \frac{n}{2^{n-1}} \frac{w^{n(n-1)/2}}{i^{n-1}} = \frac{i^{n-1}}{i^{n-1}} \frac{n}{2^{n-1}} = \frac{n}{2^{n-1}}$$

I guess (but did not check) that the same kind of reasoning gives the one with $\cos$.

Answer 3

Define $\zeta_n = e^{2 \pi i/n}$.

Proposition For odd integer $n \geq 1$, \begin{align} \prod_{k = 1}^{n-1}(\zeta_n^{k} - \zeta_n^{-k}) = n. \end{align} and \begin{align} \prod_{k = 1}^{n-1} \sin( \tfrac{2 \pi k }{n} ) = \tfrac{n}{(2 i)^{n-1}}. \end{align} Proof: The claimed identities follow from the identity \begin{align} z^n - 1 = \prod_{ k =0}^{n-1} (z - \zeta_n^{k}) = \prod_{ k =0}^{n-1} (z - \zeta_n^{-2k}). \end{align} Writing $z = x/y$, we have \begin{align} x^n - y^n = \prod_{k = 0}^{n-1} ( \zeta_n^{k} x - \zeta_n^{-k} y). \end{align} Thus, \begin{align} n y^{n-1} = \lim_{x \to y} \frac{x^n - y^n}{x - y} = \lim_{x \to y} \ \ \prod_{k = 1}^{n-1} ( \zeta_n^{k} x - \zeta_n^{-k} y) = y^{n-1} \ \prod_{k = 1}^{n-1} ( \zeta_n^{k} - \zeta_n^{-k} ). \end{align} For the second identity, let $x =e^{\pi i z}$ and $y = e^{- \pi i z}$ and recall the complex exponential representation of the sine function. This yields \begin{align} n = \lim_{z \to 0} \frac{\sin n \pi z}{\sin z } = (2 i)^{n-1} \lim_{z \to 0} \ \ \prod_{k = 1}^{n-1} \sin( \pi z + \tfrac{2 \pi k }{n} ) = (2 i)^{n-1} \prod_{k = 1}^{n-1} \sin( \tfrac{2 \pi k }{n} ). \end{align}

Similar reasoning works to prove the identities that you mention.

Answer 4

The second purported identity is equivalent to asking for the constant term of $\dfrac{U_{n-1}(x)}{2^{n-1}}$ (i.e., $\dfrac{U_{n-1}(0)}{2^{n-1}}$), where $U_n(x)$ is the Chebyshev polynomial of the second kind. Since

$$\frac{U_{n-1}(x)}{2^{n-1}}=\frac{\sin(n \arccos\,x)}{2^{n-1}\sqrt{1-x^2}}$$

letting $x=0$ gives your identity.