$\begin{array}{l} A = {\cos ^2}x + {\cos ^2}\left( {x + \dfrac{{2\pi }}{3}} \right) + {\cos ^2}\left( {\dfrac{{2\pi }}{3} - x} \right)\\ A = \dfrac{{1 + \cos 2x}}{2} + \dfrac{{1 + \cos \left( {2x + \dfrac{{4\pi }}{3}} \right)}}{2} + \dfrac{{1 + \cos \left( {\dfrac{{2\pi }}{3} - 2x} \right)}}{2}\\ A = \dfrac{{3 + \cos 2x + \cos \left( {2x + \dfrac{{4\pi }}{3}} \right) + \cos \left( {\dfrac{{4\pi }}{3} - 2x} \right)}}{2} = \dfrac{{3 + \cos 2x + 2\cos \dfrac{{4\pi }}{3}\cos 2x}}{2}\\ A = \dfrac{{3 + \cos 2x - 2\cos \dfrac{\pi }{3}\cos 2x}}{2} = \dfrac{{3 + \cos 2x - \cos 2x}}{2} = \dfrac{3}{2} \end{array}$
