Answer
$g'(\theta)=-2\sin\theta\cos\theta$ or $g'(\theta)=-\sin2\theta$
Work Step by Step
$g(\theta)=\cos^{2}\theta$
Differentiate using the chain rule:
$g'(\theta)=2\cos\theta(\cos\theta)'=2(\cos\theta)(-\sin\theta)=...$
$...=-2\sin\theta\cos\theta$
Also you can use the identity $2\sin\theta\cos\theta=\sin(2\theta)$ to present the answer like this:
$g'(\theta)=-\sin2\theta$
