1. The problem asks us to find the derivative of $y = \cos^2 x$.\n2. Rewrite $y = \cos^2 x$ as $y = (\cos x)^2$ to use the chain rule easily.\n3. Using the chain rule, the derivative $\frac{dy}{dx}$ is $2(\cos x) \times \frac{d}{dx}(\cos x)$.\n4. The derivative of $\cos x$ with respect to $x$ is $-\sin x$.\n5. Substitute this back: $\frac{dy}{dx} = 2 \cos x \times (-\sin x) = -2 \cos x \sin x$.\n6. Using the double angle identity, $\sin 2x = 2 \sin x \cos x$, so $\frac{dy}{dx} = -\sin 2x$.\n7. Therefore, the derivative of $y = \cos^2 x$ is $\boxed{-\sin 2x}$.\nAnswer choice (a) matches this derivative.