we play with sin

$\begin{aligned} \prod_{k=1}^{n-1} \sin(\frac{k \pi}{n}) &= \prod_{k=1}^{n-1} \left ( \frac{e^{\frac{i \pi k}{n}}-e^{-\frac{i \pi k}{n}}}{2i} \right ) & {\color{#D61F06} \sin(\theta) = \frac{e^{i \theta}-e^{-i\theta}}{2 i}} \\ &= \prod_{k=1}^{n-1} \left ( \frac{1}{2i} \right ) \cdot \left (- e^{-\frac{i \pi k}{n}} \right ) \cdot \left ( - e^{\frac{i 2\pi k}{n}} + 1 \right ) \\ &= \left ( \frac{-1}{2i} \right )^{n-1} \cdot e^{\frac{-i \pi}{n}\sum_{k=1}^{n-1} k} \cdot \prod_{k=1}^{n-1} \left ( 1 - e^{\frac{2\pi i}{n} k} \right ) \\ &= \frac{1}{2^{n-1} \cdot (-i)^{n-1}} \cdot e^{-\frac{i\pi}{n} \cdot \frac{(n-1)n}{2}} \cdot \prod_{k=1}^{n-1} \left ( 1 - w^k \right ) & \color{#D61F06} w \text{ is nth root of unity}\\ &= \frac{(-i)^{n-1}}{2^{n-1} \cdot (-i)^{n-1}} \cdot \lim_{x\rightarrow 1} \frac{x^n - 1}{x-1} = \frac{n}{2^{n-1}} \\ &\color{#D61F06} x^n - 1 = \prod_{k=0}^{n-1} \left ( x - w^k \right ) = (x - 1) \prod_{k=1}^{n-1} \left ( x - w^k \right ) \\ & \color{#D61F06} \Rightarrow \prod_{k=1}^{n-1} \left ( x - w^k \right ) = \frac{x^n - 1}{x-1} \end{aligned}$