\[{\text{Докажите}}{\text{, что }}\left| {\sum\limits_{k = 1}^n {\sin k} } \right| < 2.\]

\[\begin{array}{l} {\text{Вычислите:}} \hfill \\ \cos \frac{{2\pi }}{{31}}\cos \frac{{4\pi }}{{31}}\cos \frac{{8\pi }}{{31}}\cos \frac{{16\pi }}{{31}}\cos \frac{{32\pi }}{{31}}. \hfill \\ \end{array}\]

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

\[\begin{array}{l} {\text{Докажите тождества:}} \hfill \\ {\text{а) }}\sum\limits_{k = 1}^3 {\sqrt[3]{{{{\left( {\frac{{\cos \frac{{\pi k}}{7}}}{{\cos \frac{{2\pi k}}{7}}}} \right)}^2}}}} = \sqrt[3]{{49}};{\text{ б) }}\sum\limits_{k = 1}^3 {\sqrt[3]{{{{\left( {\frac{{\cos \frac{{2\pi k}}{7}}}{{\cos \frac{{\pi k}}{7}}}} \right)}^2}}}} = 2\sqrt[3]{7}; \hfill \\ {\text{в) }}\sum\limits_{k = 1}^4 {\sqrt[3]{{{{\left( {\frac{{\cos \frac{{\pi k}}{9}}}{{\cos \frac{{2\pi k}}{9}}}} \right)}^2}}}} = 2\sqrt[3]{9} + 1;{\text{ г) }}\sum\limits_{k = 1}^4 {\sqrt[3]{{{{\left( {\frac{{\cos \frac{{2\pi k}}{9}}}{{\cos \frac{{\pi k}}{9}}}} \right)}^2}}}} = 3\sqrt[3]{3} + 1. \hfill \\ \end{array}\]