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This solution herein is inspired by a comment left by @user1952009 on another posted solution. That solution is elegant in that it relies on only $(i)$ Euler’s formula to write
$$\sin(nx)=\frac{e^{inx}-e^{-inx}}{2i}\tag 1$$
and $(ii)$ the $n$ roots of unity, $z=e^{-i2k \pi/n}$ for $k=0,\dots,n-1$, of the equation $z^n=1$ to write
$$z^n-1=\prod_{k=0}^{n-1}(z-e^{-i2\pi k/n})\tag 2$$
Proceeding, we find
$$\begin{align}
\sin(nx)&=\frac{e^{inx}-e^{-inx}}{2i}&\\\\
&=\frac{e^{-inx}}{2i}\left(e^{i2nx}-1\right)\\\\
&=\frac{e^{-inx}}{2i}\prod_{k=0}^{n-1}(e^{i2x}-e^{-i2\pi k/n})\\\\
&=\frac{e^{-inx}}{2i}\prod_{k=0}^{n-1}\left(\color{blue}{e^{ix}}\color{red}{e^{-i\pi k/n}}\color{green}{(2i)}\sin\left(x+\frac{k \pi}{n}\right)\right)\\\\
&=\frac{e^{-inx}}{2i} \color{blue}{e^{inx}} \color{red}{e^{-i(\pi/n)n(n-1)/2}}\color{green}{(2i)^n}\prod_{k=0}^{n-1}\sin\left(x+\frac{k \pi}{n}\right)\\\\
&=\prod_{k=0}^{n-1}\sin\left(x+\frac{k \pi}{n}\right)
\end{align}$$
as was to be shown!
The right side is odd and periodic with period $2\pi/n$. Repeated use of the identities for $\sin(a) \sin(b)$ and $\cos(a) \sin(b)$ must give you a sum of terms of the form $a_j \sin(j x)$ and $b_j \cos(j x)$ with $0 \le j \le n$, but the oddness and periodicity implies that the terms for $j < n$ and the $\cos$ terms cancel, leaving a multiple of $\sin(nx)$. Now all you have to do is show that the constant is $1$…
You may prove such identity through Herglotz trick, for instance. Both the RHS and the LHS are entire functions in the complex plane of order $1$, having simple zeroes at every point of $\frac{\pi}{n}\mathbb{Z}$ and only there. Additionally, they both are $\frac{2\pi}{n}$-periodic functions. It follows that such identity holds as soon as it holds for some $x\not\in\frac{\pi}{n}\mathbb{Z}$, like $x=\frac{\pi}{2n}$. In such a case, it is straightforward to prove through De Moivre’s identity $\sin(x)=\frac{e^{ix}-e^{-ix}}{2}$.
Since $\sin(z)=\frac{\pi}{\Gamma(z)\Gamma(1-z)}$, you may also prove it through the multiplication theorem for the $\Gamma$ function.