三十四角形
cos  2 π 34 = cos  π 17 = 34 − 68 − 17 + 1 + 2 34 − 68 + 17 − 1 17 + 272 16 = 1 − 17 + 34 − 68 + 68 + 2448 + 2720 + 6284288 16 = 1 − 17 + 2 ⋅ 17 − 2 2 ⋅ 17 + 2 2 ⋅ 17 + 2 4 ⋅ 3 2 ⋅ 17 + 2 5 ⋅ 5 ⋅ 17 + 2 10 ⋅ 17 ⋅ 19 2 16 = 1 − 17 + 2 ⋅ 17 − 2 ⋅ 17 + 2 2 ⋅ 17 + 2 2 ⋅ 3 ⋅ 17 + 2 2 2 ⋅ 5 ⋅ 17 + 2 ⋅ 19 ⋅ 17 16 = 1 16 ( 1 − 17 + 2 ⋅ 17 − 17 + 2 ⋅ 17 + 3 17 + 2 ⋅ 5 ⋅ 17 + 19 17 ) {\displaystyle {\begin{aligned}\cos {\frac {2\pi }{34}}=\cos {\frac {\pi }{17}}=&{\frac {{\sqrt {34-{\sqrt {68}}}}-{\sqrt {17}}+1+2{\sqrt {{\sqrt {34-{\sqrt {68}}}}+{\sqrt {17}}-1}}{\sqrt {\sqrt {17+{\sqrt {272}}}}}}{16}}\\=&{\frac {1-{\sqrt {17}}+{\sqrt {34-{\sqrt {68}}}}+{\sqrt {68+{\sqrt {2448}}+{\sqrt {2720+{\sqrt {6284288}}}}}}}{16}}\\=&{\frac {1-{\sqrt {17}}+{\sqrt {2\cdot 17-{\sqrt {2^{2}\cdot 17}}}}+{\sqrt {2^{2}\cdot 17+{\sqrt {2^{4}\cdot 3^{2}\cdot 17}}+{\sqrt {2^{5}\cdot 5\cdot 17+{\sqrt {2^{10}\cdot 17\cdot 19^{2}}}}}}}}{16}}\\=&{\frac {1-{\sqrt {17}}+{\sqrt {2\cdot 17-2\cdot {\sqrt {17}}}}+{\sqrt {2^{2}\cdot 17+2^{2}\cdot 3\cdot {\sqrt {17}}+2^{2}{\sqrt {2\cdot 5\cdot 17+2\cdot 19\cdot {\sqrt {17}}}}}}}{16}}\\=&{\frac {1}{16}}\left({1-{\sqrt {17}}+{\sqrt {2}}\cdot {\sqrt {17-{\sqrt {17}}}}+2\cdot {\sqrt {17+3{\sqrt {17}}+{\sqrt {2}}\cdot {\sqrt {5\cdot 17+19{\sqrt {17}}}}}}}\right)\\\end{aligned}}}