Question

求证(2sinxcosx)/(sinx+cosx-1)(sinx-cosx+1)=(1+cosx)/sinx

Answer 1

(2sinxcosx)/(sinx+cosx-1)(sinx-cosx+1)=(1+cosx)/sinx
(2sinxcosx)/[(sinx)^2-(cosx-1)^2]=(1+cosx)/sinx
2(sinx)^2cosx=(1+cosx)(sinx)^2-(cos^x-1)(cosx-1)
2(sinx)^2cosx=(sinx)^2+cosx(sinx)^2+(sinx)^2(cosx-1)
(sinx)^2cosx=(sinx)^2cosx
得证

Answer 2

(2sinxcosx)/(sinx+cosx-1)(sinx-cosx+1)
=(2sinxcosx)/[sin^2x-(cosx-1)^2]
=(2sinxcosx)/[sin^2x-cos^2x+2cosx-1]
=(2sinxcosx)/2(cosx-cos^2x)
=sinx/(1-cosx)
=sinx(1+cosx)/(1-cosx)(1+cosx)
=sinx(1+cosx)/sin^2x
=(1+cosx)/sinx

Answer 3

证明：左边=2sinxcosx/[sin^2x-(cosx-1)^2]

=2sinxcosx/(sin^2x-cos^2x+2cosx-1)

=2sinxcosx/(2cosx-2cos^2x)

=sinx/(1-cosx)

=[(1-cos^2x)/sinx]/(1-cosx)

=(1+cosx)/sinx

证毕