已知sin^4 α/cos^2 β+cos^4 α/sin^2 β=1,求证sin^4 β/cos^2 α+cos^4

证明：右柯西不等式可知
1=sin^4 α/cos^2 β+cos^4 α/sin^2 β
=(cos^2β+sin^2β)(sin^4 α/cos^2 β+cos^4 α/sin^2 β)
≥(sin^2α+cos^2α)^2=1
当且仅当（sin^4 α/cos^2 β)/(cos^2β)=(cos^4 α/sin^2 β)/(sin^2β)时等号成立,即
sin^2α/cos^2α=cos^2 β/sin^2 β
所以sin^4 β/cos^2 α+cos^4 β/sin^2 α
=cos^2βsin^2β/sin^2α+cos^2βsin^2β/cos^2 α
=cos^2βsin^2β(1/sin^2α+1/cos^2 α)
=cos^2βsin^2β/(sin^2αcos^2 α)
=1
所以sin^4 β/cos^2 α+cos^4 β/sin^2 α=1