Question

求证：[（sinx+cosx-1）（sinx-cosx+1）]/sin2x=(1-cosx)/sinx

Answer 1

[（sinx+cosx-1）（sinx-cosx+1）]/sin2x
=[sinx+(cosx-1)][sinx-(cosx-1)]/sin2x
=[(sinx)^2-(cosx-1)^2]/sin2x
=[(sinx)^2-(cosx)^2+2cosx-1]/sin2x
=[(sinx)^2-(cosx)^2+2cosx-1]/sin2x
=-[-(sinx)^2+(cosx)^2-2cosx+1]/sin2x
=-[1-(sinx)^2+(cosx)^2-2cosx]/sin2x
=-[(cosx)^2+(cosx)^2-2cosx]/sin2x
=-[2(cosx)^2-2cosx]/2sinxcosx
=-2cosx[cosx-1]/2sinxcosx
=-[cosx-1]/sinx
=(1-cosx)/sinx