IAL 2021 Oct Q5

Given that $y = \tan^2 x$

(a) show that

where $p$ and $q$ are integers to be determined.

(b) Hence determine the Taylor series expansion about $\frac{\pi}{3}$ of $\tan^2 x$ in ascending powers of $\left( x - \frac{\pi}{3} \right)$ up to and including the term in $\left( x - \frac{\pi}{3} \right)^3$ , giving each coefficient in simplest form.