Prove that cos π/7 - cos 2π/7 + cos 3π/7 = 1/2.
Solution
Consider the roots of x7 + 1 = 0. They are eiπ/7, ei3π/7, ... , ei13π/7 and must have sum zero since there is no x6 term. Hence, in particular, their real parts sum to zero. But cos7π/7 = - 1 and the others are equal in pairs, because cos(2π - x) = cos x. So we get cos π/7 + cos 3π/7 + cos 5π/7 = 1/2. Finally since cos(π - x) = - cos x, cos 5π/7 = - cos 2π/7.
Solutions are also available in: Samuel L Greitzer, International Mathematical Olympiads 1959-1977, MAA 1978, and in István Reiman, International Mathematical Olympiad 1959-1999, ISBN 189-8855-48-X.
© John Scholes
jscholes@kalva.demon.co.uk
21 Sep 1998
Last corrected/updated 24 Sep 2003