De Moivre's theorem and series in complex numbers

I came across this question from an old past paper where no solutions were provided; I only need guidance and I will give it a go...

Given that $$w_n=3^{(-n)} \cos⁡2nθ$$ [(for n=1,2,3….〗Use De Moivre’s theorem to show that $$1 + w_{1} + w_2 + \cdots + w_{(n-1)} = \frac{(9 - 3 \cos2\theta + 3^{(-n-1)} \cos 2⁡(n-1)\theta-3^{(-n-1)} \cos2n\theta)}{(10 - 6 \cos 2\theta)}$$

Hint: $$w_n$$ is the real part of the complex number $$z_n = (3)^{-n} (\cos(2n \theta) + i \sin(2n \theta))$$. So $$1 + w_1 \ldots + w_{n-1}$$ is the real part of $$1 + z_1 + \ldots + z_{n-1}$$.
• It wasn't what I had in mind but it may help. If you let $a = \frac{1}{3} (\cos(2\theta) + i \sin(2\theta))$, how does $1 + z_1 + \ldots + z_{n-1}$ look ?