Differentiate a tangent expansion to obtain a secant-square series

Let $y=\tan 2x$, show that

(i) $dy/dx=2(1+y^2)$,

(ii) $d^3y/dx^3=4(dy/dx)^2+4y\,d^2y/dx^2$.

Find the Maclaurin series for $\tan 2x$, up to and including the term in $x^3$.

Hence, find the first two nonzero terms of the Maclaurin series for $\sec^2 2x$.