试用分析法证明:(1+1/sin^2θ)(1+1/cos^2θ)≥9
问题描述：
试用分析法证明:(1+1/sin^2θ)(1+1/cos^2θ)≥9
答
(1+1/sin^2θ)(1+1/cos^2θ)
=1+1/sin^2θ+1/cos^2θ+1/sin^2θcos^2θ
=1+(sin^2θ+cos^2θ)/sin^2θcos^2θ+1/sin^2θcos^2θ
=1+2/sin^2θcos^2θ
=1+8/4sin^2θcos^2θ
=1+8/(2sinθcosθ)^2
=1+8/sin^2 2θ
因为0=1+8=9