Q:

If $a_k=\cos(2k\pi/n)-2+i\sin(2k\pi/n)$, evaluate $a_1a_2\cdots a_n$

Define
$$a_k=\cos\frac{2k\pi}{n}-2+i\sin\frac{2k\pi}{n}$$
How can I approach this product?
$$a_1 a_2 \cdots a_n$$
I tried to investigate if one of terms is $0$, as it is a product, but no.
Any ideas?

I have already calculated this sum using the Euler formula.
$$a_1+a_2+\cdots +a_n,\;n>1$$
(original problem images: $a_k$, product, sum)

A:

So, $a_k;1\le k \le n$
are the roots of $$(x+2)^n=1\iff x^n+\cdots+2^n-1=0$$
Using Vieta's formula $$\prod_{k=1}^n a_k=(-1)^n\cdot\dfrac{2^n-1}1$$

