1. The problem is to verify the identity $\cos^2(x) = 1$. 2. Recall the Pythagorean identity: $$\sin^2(x) + \cos^2(x) = 1$$ which holds for all real $x$. 3. From this identity, $\cos^2(x) = 1 - \sin^2(x)$. 4. The equation $\cos^2(x) = 1$ implies $1 - \sin^2(x) = 1$. 5. Simplifying, $\sin^2(x) = 0$. 6. This means $\sin(x) = 0$. 7. The solutions to $\sin(x) = 0$ are $x = n\pi$ where $n$ is any integer. 8. Therefore, $\cos^2(x) = 1$ only when $x = n\pi$ for integers $n$. In summary, $\cos^2(x) = 1$ is not true for all $x$, but only at specific points where $\sin(x) = 0$. Final answer: $\cos^2(x) = 1$ if and only if $x = n\pi$, $n \in \mathbb{Z}$.