1. The problem is to simplify the expression $\cos 80^\circ \cos 20^\circ + \sin 80^\circ \sin 20^\circ$.\n\n2. We use the cosine addition formula: $$\cos(A - B) = \cos A \cos B + \sin A \sin B$$\nThis formula states that the cosine of the difference of two angles equals the sum of the product of their cosines and the product of their sines.\n\n3. Comparing the given expression to the formula, we identify $A = 80^\circ$ and $B = 20^\circ$.\n\n4. Substitute into the formula: $$\cos 80^\circ \cos 20^\circ + \sin 80^\circ \sin 20^\circ = \cos(80^\circ - 20^\circ)$$\n\n5. Simplify the angle difference: $$80^\circ - 20^\circ = 60^\circ$$\n\n6. Therefore, the expression simplifies to: $$\cos 60^\circ$$\n\n7. Recall that $\cos 60^\circ = \frac{1}{2}$.\n\nFinal answer: $$\frac{1}{2}$$