1. The problem is to simplify or evaluate the expression $\cos(x)^2 - 0.5$. 2. Recall the Pythagorean identity: $$\cos^2(x) + \sin^2(x) = 1$$ and the double-angle formula for cosine: $$\cos(2x) = 2\cos^2(x) - 1$$. 3. From the double-angle formula, solve for $\cos^2(x)$: $$\cos(2x) = 2\cos^2(x) - 1 \implies 2\cos^2(x) = \cos(2x) + 1 \implies \cos^2(x) = \frac{\cos(2x) + 1}{2}$$ 4. Substitute this into the original expression: $$\cos^2(x) - 0.5 = \frac{\cos(2x) + 1}{2} - \frac{1}{2}$$ 5. Simplify the right side: $$= \frac{\cos(2x) + 1 - 1}{2} = \frac{\cos(2x)}{2}$$ 6. Therefore, the expression simplifies to: $$\boxed{\frac{\cos(2x)}{2}}$$ This means $\cos(x)^2 - 0.5$ is equivalent to half of $\cos(2x)$.