近期在阅读贵刊 2 0 0 2年第 3期时 ,看到了安徽师范大学的郭老师给出的《半角的余弦和上界的加强》一文 ,觉得证明较繁 .实际上利用柯西不等式结合恒等式 cos A +cos B+cos C =1+rR证明较为简便 .现证如下 :cos A2 +cos B2 +cos C2　≤ 3(cos2 A2 +cos2 B2 +cos2 C2 )　 = 32 In the recent reading of the third issue of your magazine in 2002, you saw the article “Enhancement of the cosine of the half angle and the reinforcement of the upper bound” given by Prof. Guo from Anhui Normal University. I feel that the proof is more complicated. In fact, we use the Cauchy inequality to combine The identity cos A + cos B + cos C = 1 + rR proves relatively simple. The proof is as follows: cos A2 + cos B2 + cos C2 ≤ 3 (cos2 A2 + cos2 B2 + cos2 C2) = 32