2020 NYJC J2 CT P1 Q6

Show that \(\tan 2\theta =\frac{2\tan \theta }{1-{{\tan }^{2}}\theta }\) using the addition formula from the List of Formulae (MF26).[1]

By using the substitution \(x=\tan \theta \), or otherwise, find the exact value of  \(\displaystyle\displaystyle \displaystyle\int_{0}^{1}{{{\tan }^{-1}}\left( \frac{2x}{1-{{x}^{2}}} \right)}\,\mathrm{d}x\).[5]