Calculate $Re$ and $Im$ of $(1+w)\cdot(1+w^2)\cdot(1+w^3)\cdots(1+w^{100})$

Question

Given $w \in \mathbb{C}$, $w = e^{i\frac{2\pi}{3}}$. Calculate $Re$ and $Im$ of $(1+w)\cdot(1+w^2)\cdot(1+w^3)\cdots(1+w^{100})$.

$(\frac{1}{2} + i \frac{\sqrt{3}}{2})(\frac{1}{2} - i \frac{\sqrt{3}}{2})\cdot2\cdot(\frac{1}{2} + i \frac{\sqrt{3}}{2})(\frac{1}{2} - i \frac{\sqrt{3}}{2})\cdot2\cdots(\frac{1}{2} + i \frac{\sqrt{3}}{2}) = ((\frac{1}{2} + i \frac{\sqrt{3}}{2})(\frac{1}{2} - i \frac{\sqrt{3}}{2})\cdot2)^{33}\cdot(\frac{1}{2} + i \frac{\sqrt{3}}{2})$

How to finish it?

Answer 1

Hint:

$$\left(\frac12+i\frac{\sqrt 3}2\right)\left(\frac12-i\frac{\sqrt 3}2\right)=\frac14+\frac34=1$$