Problem involving roots of unity

Question

Let $\varepsilon _k$ be $\cos \frac {2k \pi} {n} + i \sin \frac {2k \pi} {n}$. Find the value of the product: $$\prod _ {k=1}^n (2+\varepsilon _k-\varepsilon _k^2).$$

Answer 1

For $z^n-1=0$ : $$ \prod_1^n(2+\varepsilon_k-\varepsilon_k^2)=\prod_1^n-(-1-\varepsilon_k)(2-\varepsilon_k)=(-1)^n \prod_1^n(-1-\varepsilon_k)\prod_1^n(2-\varepsilon_k)=(-1)^n((-1)^n-1)(2^n-1) $$