Abstract: Let n be a positive integer and  p=4n+1 be a prime.The author proves that there exist n positive integers ki(i=1,2,…,n) which satisfy ki≤2n and ｎｉ＝１ｃｏｓ(２ｋｉ－１)/ｐ＝（１＋ｐ）／４.

Key words: odd prime, trigonometric sum, Legendre symbol