已知tanx+sinx=a,tanx-sinx=b.求证(a^2-b^2)^2=16ab
问题描述：
已知tanx+sinx=a,tanx-sinx=b.求证(a^2-b^2)^2=16ab
答
(a^2-b^2)^2-16ab
=[(a-b)(a+b)]^2-16ab
=(4sinxtanx)^2-16(tan^2x-sin^2x)
=16(sin^2xtan^2x-tan^2x+sin^2x)
=16[(sin^2x-1)(tan^2x+1)+1]
=16(-cos^2xcos^2x+1)
=0
∴(a^2-b^2)^2=16ab
其中sin^2x,tan^2x,cos^2x为sinx,tanx,cosx的平方