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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

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

Yes, that is correct. The left side (LS) of the equation, which is (sinx - cosx)^2, simplifies to 1 - 2sinxcosx. The right side (RS) of the equation is also 1 - 2sinxcosx. Since both sides are equal, LS = RS.