Help With Complex Number Sequence

Question

I have the $n$ complex numbers $z_1 , z_2 , z_3 , \dots , z_n$, such that $|z_k| = 1 $ for any $1\leq k\leq n$ and $z_1 \cdot z_2 \cdot z_3 \cdots \cdot z_n = 1$. I need to prove that $$(1+z_1) \cdot (1 + z_2) \cdots (1 + z_n)$$ is a real number.

Answer 1

Let $w:=\prod_k(1+z_k)$ so $w^\ast=\prod_k(1+z_k^\ast)=\prod_k(1+z_k^{-1})=w/\prod_kz_k=w$. Since $w^\ast=w$, $w\in\Bbb R$.