Answer 1

We have:

\[\tan px = \cot qx\]

\[\Rightarrow \tan px = \tan \left( \frac{\pi}{2} - qx \right)\]

\[ \Rightarrow px = n\pi + \left( \frac{\pi}{2} - qx \right), n \in Z\]

\[ \Rightarrow (p + q)x = n\pi + \frac{\pi}{2}, n \in Z\]

\[ \Rightarrow x = \left( \frac{2n + 1}{p + q} \right)\frac{\pi}{2}, n \in Z\]