$$\begin{array}{l}\cos \frac{\pi }{{17}} \cdot \cos \frac{{4\pi }}{{17}} = \frac{{1 + \sqrt {17} + \sqrt {34 + 2\sqrt {17} } }}{{16}} \hfill \\ \frac{1}{{\cos \frac{\pi }{{17}}}} + \frac{1}{{\cos \frac{{2\pi }}{{17}}}} - \frac{1}{{\cos \frac{{4\pi }}{{17}}}} + \frac{1}{{\cos \frac{{8\pi }}{{17}}}} = 2\sqrt {17 + 4\sqrt {17} }\end{array} $$