化简(sin x+cos x-1)(sin x-cos x+1)/sin 2x

来源：学生作业帮助网编辑：作业帮时间：2024/04/28 00:32:46

化简(sin x+cos x-1)(sin x-cos x+1)/sin 2x
化简(sin x+cos x-1)(sin x-cos x+1)/sin 2x

化简(sin x+cos x-1)(sin x-cos x+1)/sin 2x
sinx=2sin(x/2)cos(x/2)
cosx=1-2sin^2(x/2)
故sin x+cos x-1=2sin(x/2)[cos(x/2)-sin(x/2)]
sin x-cos x+1=2sin(x/2)[cos(x/2)+sin(x/2)]
(sin x+cos x-1)(sin x-cos x+1)=4sin^2(x/2)[cos^2(x/2)-sin^2(x/2)]
=4sin^2(x/2)cosx
sin2x=2sinxcosx
(sin x+cos x-1)(sin x-cos x+1)/sin 2x
=4sin^2(x/2)cosx/2sinxcosx
=2sin^2(x/2)/sinx
=sin^2(x/2)/[sin(x/2)cos(x/2)]
=tan(x/2)

(sin x+cos x-1)(sin x-cos x+1)/sin 2x
=(sin x+cos x-1)(sin x-(cos x-1))/sin 2x
=[sin²x-(cos x-1)²]/sin2x
=[sin²x-cos²x+2cosx-1]/sin2x
=[-cos²x+2cosx-cos²x]/sin2x
=2cosx(1-cosx)/2sinxcosx
=(1-cosx)/sinx
=(1-(1-2sin²(x/2))/2sin(x/2)cos(x/2)
=tan(x/2)

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