<s>[INST] Prove trigonometric relation Recently, I found this identities in a sheet of paper I was given as studying material:
$$\prod^n_{k=1}\sin\left(\frac{k\pi}{2n+1}\right)=\frac{\sqrt{2n+1}}{2^n}\tag1$$
$$\prod^n_{k=1}\cos\left(\frac{k\pi}{2n+1}\right)=\frac{1}{2^n}\tag2$$
$$\prod^n_{k=1}\tan\left(\frac{k\pi}{2n+1}\right)=\sqrt{2n+1}\tag3$$
They were there as auxiliary identites, meaning that we could use them if we needed to. But I found them really interesting, so I tried to prove them. After some time struggling with equations, I managed to prove $(2)$ for $n=1,2,3$, with increasing difficulty using the sum of cosines formula, the product of cosines, and the formula for the arithmetic progression of cosines. I couldn't finish the proof for $n=4$, because I needed that:
$$\left(2\cos\frac{\pi}{9}\right)\left(2\cos\frac{2\pi}{9}-1\right)=1\tag{4}$$
Equivalently, showing that $\cos\frac{\pi}{9}$ is a solution of
$$8x^3-2x-1=0$$
And no idea with $(1)$ and $(3)$.The best I could think of is that since the square root can't appear out of the blue, considering the square and doing a clever trick that results in a sum that yields $2n+1$ in some way would do it. But I have had no luck until now. Any ideas?
Edit: Actually, the proof for $(4)$ wasn't that hard:
$$2*2\cos\frac{\pi}{9}\cos\frac{2\pi}{9}-2\cos\frac{\pi}{9}=1$$
$$\iff 2*\left(\cos\frac{\pi}{3}+\cos\frac{\pi}{9}\right)-2\cos\frac{\pi}{9}=1$$
$$\iff 2*\left(\frac{1}{2}+\cos\frac{\pi}{9}\right)-2\cos\frac{\pi}{9}=1$$
The last one is clearly true.
 [/INST] Consider
$$\prod_{k=1}^{2n} \sin \left(\frac{k\pi}{2n+1}\right).$$
By symmetry, it's the square of your first product, and expanding it with Euler's formula, we find
$$\begin{align}
\prod_{k=1}^{2n} \sin \left(\frac{k\pi}{2n+1}\right) &= \frac{1}{(2i)^{2n}}\prod_{k=1}^{2n} \left(\exp \frac{k\pi i}{2n+1} - \exp \frac{-k\pi i}{2n+1}\right)\\
&= \frac{1}{(2i)^{2n}} \exp\left(\frac{\pi i (2n)(2n+1)}{2(2n+1)}\right) \prod_{k=1}^{2n}\left(1-\exp \frac{-2k\pi i}{2n+1}\right)\\
&= \frac{1}{2^{2n}}\prod_{k=1}^{2n}\left(1-\exp \frac{-2k\pi i}{2n+1}\right).
\end{align}$$
Now consider the polynomial
$$P(X) = \prod_{k=1}^{2n} \left(X - \exp \frac{-2k\pi i}{2n+1}\right).$$
All its roots are $2n+1$-th roots of unity other than $1$, and they are all distinct, since
$$\frac{-k}{2n+1} - \frac{-j}{2n+1} \notin \mathbb{Z}$$
for $k\neq j$ and $1 \leqslant k,j \leqslant 2n$. So we have
$$P(X) = \frac{X^{2n+1}-1}{X-1} = X^{2n} + X^{2n-1} + \dotsc + X + 1,$$
and we see that $P(1) = 2n+1$, which proves the first equation.
The product of the cosines is similar, we get
$$\prod_{k=1}^{2n} \cos \frac{k\pi}{2n+1} = \frac{(-1)^n}{2^{2n}} \prod_{k=1}^{2n}\left(1 + \exp \frac{-2k\pi i}{2n+1}\right).$$
Considering
$$Q(X) = \prod_{k=1}^{2n}\left(X + \exp \frac{-2k\pi i}{2n+1}\right) = P(-X),$$
we find $Q(1) = 1$, so
$$\prod_{k=1}^{2n} \cos \frac{k\pi}{2n+1} = \frac{(-1)^n}{2^{2n}}.$$
Since the factor for $k = 2n+1-j$ is the negative of the factor for $j$, $1 \leqslant j \leqslant n$, we have
$$\prod_{k=1}^n \cos \frac{k\pi}{2n+1} = \frac{1}{2^n}.$$
The formula for the product of the tangents is a direct consequence.
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