N2016 P1 Q8

It is given that \(y=\mathrm{f}\left( x \right)\), where \(\mathrm{f}\left( x \right)=\tan \left( ax+b \right)\) for constants \(a\) and \(b\).

1. Show that \(\mathrm{{f}'}\left( x \right)=a+a{{y}^{2}}\). Use this result to find \(\mathrm{{f}''}\left( x \right)\) and \(\mathrm{{f}'''}\left( x \right)\) in terms of \(a\) and \(y\).[5]
2. In the case where \(b=\frac{1}{4}\pi \), use your results from part (i) to find the Maclaurin series for \(\mathrm{f}\left( x \right)\) in terms of \(a\), up to and including the term in \({{x}^{3}}\).[3]
3. Find the first two non-zero terms in the Maclaurin series for \(\tan 2x\).[3]