that is correct, but it would have been simpler when you got the LS expanded, to see that you had
LS = sin^2x - 2sinxcosx + cos^2x
= sin^2x + cos^2x -2sinxcosx
= 1 - 2sinxcosx
Is this correct?
(sinx - cosx)^2 = 1 - 2sinxcosx
LS = sin^2x - 2sinxcosx + cos^2x
= 1 - cos^2x - 2sinxcosx + cos^2x
= 1 - cos^2x + cos^2x - 2sinxcosx
= 1 - 2sinxcosx
LS = RS
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