Consider the following for the next items that follow:

Let $\rm p=\cos \left(\frac{\pi}{5}\right) \cos \left(\frac{2 \pi}{5}\right) $ and $\rm q=\cos \left(\frac{4 \pi}{5}\right) \cos \left(\frac{8 \pi}{5}\right)$.

What is the value of p + q?

Consider the following for the next items that follow:

Let $\rm p=\cos \left(\frac{\pi}{5}\right) \cos \left(\frac{2 \pi}{5}\right) $ and $\rm q=\cos \left(\frac{4 \pi}{5}\right) \cos \left(\frac{8 \pi}{5}\right)$.

What is the value of pq?

Consider the following for the next items that follow:

Let $p=\frac{1}{3}-\frac{\tan 3 x}{\tan x}$ and q = 1 - 3 tan2 x, 0 < x < π, $ x \neq \frac{\pi}{2}$.

What is pq equal to ?

Consider the following for the next items that follow:

Let $p=\frac{1}{3}-\frac{\tan 3 x}{\tan x}$ and q = 1 - 3 tan2 x, 0 < x < π, $ x \neq \frac{\pi}{2}$.

For how many values of x does $\frac{1}{p}$ become zero?