二十一角形
2 cos  14 π 21 = 2 cos  2 π 3 = − 1 2 cos  6 π 21 + 2 cos  12 π 21 + 2 cos  18 π 21 = 2 cos  2 π 7 + 2 cos  4 π 7 + 2 cos  6 π 7 = − 1 {\displaystyle {\begin{aligned}&2\cos {\frac {14\pi }{21}}=2\cos {\frac {2\pi }{3}}=-1\,\\&2\cos {\frac {6\pi }{21}}+2\cos {\frac {12\pi }{21}}+2\cos {\frac {18\pi }{21}}=2\cos {\frac {2\pi }{7}}+2\cos {\frac {4\pi }{7}}+2\cos {\frac {6\pi }{7}}=-1\\\end{aligned}}}
整理すると
2 cos  2 π 21 + 2 cos  4 π 21 + 2 cos  8 π 21 + 2 cos  10 π 21 + 2 cos  16 π 21 + 2 cos  20 π 21 = 1 {\displaystyle 2\cos {\frac {2\pi }{21}}+2\cos {\frac {4\pi }{21}}+2\cos {\frac {8\pi }{21}}+2\cos {\frac {10\pi }{21}}+2\cos {\frac {16\pi }{21}}+2\cos {\frac {20\pi }{21}}=1}