trigonometry - Product of projections of equispaced rotating vector

When equal and equi-spaced forces are summed on y-axis what is vector sum? How do we derive the formula

$$\sum_{k=1}^{n-1}\sin\frac{\pi k}{n} = \cot \frac{\pi}{2 n}$$

( Formula given by Marco Cantarini in comments below. )

By a similar token, can

$$\prod_{k=1}^{n-1}\sin\frac{\pi k}{n}=\frac{2n}{2^n}$$

represent some physics force multiplication situation or any generalized law in which

this analogue is valid? (Formula mentioned by Jack D'Aurizio in a recent thread

Geometric proof of $\frac{\sin{60^\circ}}{\sin{40^\circ}...}$).

Answer

With some preliminary manipulations, both the identities can be derived by regarding

$$\zeta_k = \sin\frac{\pi k}{n}$$
as roots of a suitable Chebyshev polynomial, then applying Vieta's formulas - relations between the roots and the coefficients of a monic polynomial.