$\cos ^{2}\left(A-120^{\circ}\right)+\cos ^{2} A +\cos ^{2}\left(A+120^{\circ}\right) = ?$

Solve:$\begin{array}{l} \cos ^{2}\left(A-120^{\circ}\right)+\cos ^{2} A +\cos ^{2}\left(A+120^{\circ}\right) \\ =\frac{1}{2}\left\{1+\cos 2\left(A-120^{\circ}\right)+1+\cos 2 A+1\right. \\ \left.+\cos 2\left(A^{A}+120^{\circ}\right)\right\} \\ =\frac{1}{2}\left\{3+\cos \left(2 A-240^{\circ}\right)\right. \\ \left.+\cos \left(2 A+240^{\circ}\right)+\cos 2 A\right\} \\ =\frac{1}{2}\left\{3+2 \cos 2 A \cos 240^{\circ}+\cos 2 A\right\} \end{array}$

$\begin{array}{l} =\frac{1}{2}\left\{3+2 \cos 2 A \cos \left(180^{\circ}+60^{\circ}\right)+\cos 2 A\right\} \\ =\frac{1}{2}\left\{3+2 \cos 2 A\left(-\cos 60^{\circ}\right)+\cos 2 A\right\} \\ =\frac{1}{2}\left\{3+2 \cdot \cos 2 A\left(-\frac{1}{2}\right)+\cos 2 A\right\} \\ =\frac{1}{2}(3-\cos 2 A+\cos 2 A)=\frac{3}{2} \end{array}$