So I came across the following theorem:

If the interpolation node are Chebyshev points of the second kind given by :
$$ x\_k=\cos \left( \frac{j\pi}{n}\right) \qquad ( 0 \leq j \leq n) $$

Then the barycetric coefficients are :

$\beta\_j=(-1)^j\theta\_j $

Where $\theta\_j=\frac{1}{2}$ when $j=0$ or $j=n$ and $\theta\_j=1$ in other cases.

But how come? I can't seem to come across the proof of this, and I can't seem to get this coefficients by calculating $\beta\_i=1/\prod\limits\_{j=0,j\neq i}^{j=n} \left( \cos \left( \frac{i\pi}{n}\right) - \cos \left( \frac{j\pi}{n}\right)\right)$