求证1/sin^2a+3/cos^2a>=4+2根号下3
问题描述：
求证1/sin^2a+3/cos^2a>=4+2根号下3
1/sin²a +3/cos²a
=(sin²a+cos²a)/sin²a +3(sin²a+cos²a)/cos²a
=1+cos²a/sin²a+3+3sin²a/cos²a
=4+cos²a/sin²a+3sin²a/cos²a
≥4+2√[(cos²a/sin²a)(3sin²a/cos²a)]=4+2√3
答
1 /sin²a +3 /cos²a
=csc²a +3sec²a
=cot²a+1 +3 (tan²a+1)
=4+cot²a+3tan²a≥4 +2√(cot²a*3tan²a)=4 +2√3.