Malvina Baica

Copyright © 1986 Malvina Baica. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper the author obtains new trigonometric identities of the form 2(p−1)(p−2)2∏k=1p−2(1−cos2πkp)p−1−k=pp−2 which are derived as a result of relations in a cyclotomic field ℛ(ρ), where ℛ is the field of rationals and ρ is a root of unity.

Those identities hold for every positive integer p≥3 and any proof avoiding cyclotomic fields could be very difficult, if not insoluble. Two formulas∑k=1p−12(−1)(p2k)tanp−1−2kϕ=0  and−1+∑k=0p−12(−1)k(∑i=0p−1−2k2(p2k+2i)(k+1k))cosp−2kϕ=0stated only by Gauss in a slightly different form without a proof, are obtained and used in this paper in order to give some numeric applications of our new trigonometric identities.