1. The problem is to find the derivative of the function $f(x) = \cos^2(x)$. 2. We recognize that $\cos^2(x)$ means $(\cos(x))^2$, so we will use the chain rule for differentiation. 3. The chain rule states that if $y = (u(x))^n$, then $\frac{dy}{dx} = n(u(x))^{n-1} \cdot \frac{du}{dx}$. 4. Here, $u(x) = \cos(x)$ and $n = 2$, so: $$\frac{d}{dx} \cos^2(x) = 2 \cos(x) \cdot \frac{d}{dx} \cos(x)$$ 5. The derivative of $\cos(x)$ is $-\sin(x)$, so: $$\frac{d}{dx} \cos^2(x) = 2 \cos(x) \cdot (-\sin(x)) = -2 \cos(x) \sin(x)$$ 6. Therefore, the derivative of $\cos^2(x)$ is: $$\boxed{-2 \cos(x) \sin(x)}$$ This result can also be expressed using the double-angle identity $\sin(2x) = 2 \sin(x) \cos(x)$, so: $$-2 \cos(x) \sin(x) = -\sin(2x)$$ Hence, an equivalent form is: $$\frac{d}{dx} \cos^2(x) = -\sin(2x)$$