Use the printed half-unit initial value in a derivative recurrence

Given $f^{\prime}(x)=f(x)\ln(\sec(ax))$ and $f(0)=1/2$, where $a$ is real and nonzero, show

$$f^{\prime\prime}(x)f(x)=[f^{\prime}(x)]^2+a[f(x)]^2\tan(ax).$$

Hence find the Maclaurin series through $x^3$ in terms of $a$.4 marks