$$\sqrt[3]{{{{\left( {\frac{{\operatorname{tg} \frac{{2\pi }}{7}}}{{\operatorname{tg} \frac{\pi }{7}}}} \right)}^2}}} + \sqrt[3]{{{{\left( {\frac{{\operatorname{tg} \frac{{4\pi }}{7}}}{{\operatorname{tg} \frac{{2\pi }}{7}}}} \right)}^2}}} + \sqrt[3]{{{{\left( {\frac{{\operatorname{tg} \frac{{6\pi }}{7}}}{{\operatorname{tg} \frac{{3\pi }}{7}}}} \right)}^2}}} = \frac{4}{{\sqrt[3]{{3\sqrt[3]{7} - 5}}}}$$