\[\sqrt[3]{{{{\cos }^2}\frac{\pi }{7}}} + \sqrt[3]{{{{\cos }^2}\frac{{2\pi }}{7}}} + \sqrt[3]{{{{\cos }^2}\frac{{3\pi }}{7}}} = \sqrt[3]{{\frac{{6\sqrt[3]{7} + 3{{\sqrt[3]{7}}^2} + 11}}{4}}}.\]