1. Let's clarify the difference between $\cos 2^\circ$ and $\cos^2 1^\circ$.\n 2. The expression $\cos 2^\circ$ means the cosine of $2$ degrees. It's simply the cosine function applied to $2$ degrees.\n 3. The expression $\cos^2 1^\circ$ means $\left(\cos 1^\circ\right)^2$, which is the square of the cosine of $1$ degree.\n 4. These two expressions are not the same:\n - $\cos 2^\circ$ is $\cos(2)$\n- $\cos^2 1^\circ$ is $\left(\cos(1)\right)^2$\n 5. To confirm they differ, recall the double angle identity for cosine:\n $$\cos 2\theta = 2\cos^2 \theta - 1$$\n If we let $\theta = 1^\circ$, then:\n $$\cos 2^\circ = 2\cos^2 1^\circ - 1$$\n which shows $\cos 2^\circ$ and $\cos^2 1^\circ$ are related but not equal.\n 6. Therefore, it is incorrect to say $\cos 2^\circ$ equals $\cos^2 1^\circ$.\n Final answer: $\cos 2^\circ \neq \cos^2 1^\circ$.