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.