证明：cosA+cosB=2cos[(A+B)/2]*cos[(A-B)/2] =2sin(C/2)cos[(A-B)/2] ≤2sin(C/2)cosA+cosB =(b²+c²-a²)/2bc+ (a²+c²-b²)/2ac =(ab²+ac²+a²b+bc²-a³-b³)/2abc =(a+b)(2ab+c²-a²-b²)/2abc ≤(a+b)c/2abcos²A+cos²B =1+(cos2A+cos2B)/2 =1+cos(A+B)cos(A-B) =1-cosCcos(A-B) ≥1-cosC1/(1＋cos²A＋cos²B)＋1/(1＋cos²B＋ cos²C)＋1/(1＋cos²C＋cos²A)≤2 ≤1/(2-cosC)＋1/(2-cosA)＋1/(2－cosB) 不等式当且仅当A=B=C=60°时，取等号