To find the value of $\cos^2(180^\circ) - \sin^2(180^\circ)$, we can use the double-angle identity for cosine:

$\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$

Setting $\theta = 180^\circ$, we have:

$\cos(2 \cdot 180^\circ) = \cos^2(180^\circ) - \sin^2(180^\circ)$

Calculating $\cos(360^\circ)$:

$\cos(360^\circ) = 1$

Next, we can find the values of $\cos(180^\circ)$ and $\sin(180^\circ)$:

$\cos(180^\circ) = -1 \quad \text{and} \quad \sin(180^\circ) = 0$

Now substituting these into the identity:

$\cos^2(180^\circ) - \sin^2(180^\circ) = (-1)^2 - (0)^2 = 1 - 0 = 1$

Thus, the exact value of $\cos^2(180^\circ) - \sin^2(180^\circ)$ is:

$\boxed{1}$

Do you want any details or have any questions? Here are some related questions you might consider:

1. What are other double-angle identities?
2. How do you calculate cosine and sine for other angles?
3. Can you derive the double-angle identity for sine?
4. What is the unit circle and how does it relate to these calculations?
5. How would you apply these identities in solving more complex trigonometric equations?

Tip: Remember that knowing the basic angles (0°, 30°, 45°, 60°, 90°, etc.) and their sine and cosine values helps simplify many trigonometric problems.