Step 1: Test the isosceles case:
The expression is a constant for every triangle with $C = 60^{\circ}$, so pick the equilateral triangle: $A = B = 60^{\circ}$.
Step 2: Evaluate:
$\cos^2 60^{\circ} + \cos^2 60^{\circ} + \cos 60^{\circ}\cos 60^{\circ} = \frac14 + \frac14 + \frac14 = \frac34$.
The general proof by squaring the sum formula gives the same constant, so the answer is $\frac34$.
Final Answer:
The value is $\frac{3}{4}$, option (A).
\[ \boxed{\frac{3}{4}} \]