(a)[2]
If $y = \sec^3\theta$ is rewritten as $\frac{1}{\cos^3\theta}$, demonstrate that $\frac{dy}{d\theta} = 3\sin\theta\sec^4\theta$.
(b)[8]
The variables $x$ and $\theta$ obey the differential equation $(x^2 + 9)\sin\theta\,\frac{d\theta}{dx} = (x + 3)\cos^4\theta$. It is given that $x = 3$ when $\theta = \frac{\pi}{3}$. Solve the differential equation to determine the value of $\cos\theta$ when $x = 0$. Give your answer correct to 3 significant figures.