Trigonometric identities

In this paper the author obtains new trigonometric identities of the form 2 ( p − 1 ) ( p − 2 ) 2 ∏ k = 1 p − 2 ( 1 − cos 2 π k p ) p − 1 − k = p p − 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 = 1 p − 1 2 ( − 1 ) ( p 2 k ) tan p − 1 − 2 k ϕ = 0 and − 1 + ∑ k = 0 p − 1 2 ( − 1 ) k ( ∑ i = 0 p − 1 − 2 k 2 ( p 2 k + 2 i ) ( k + 1 k ) ) cos p − 2 k ϕ = 0 stated 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.