Prove the identity 1 cos x sin x2 sin2xSolution1cos x si

Prove the identity 1 – (cos x – sin x)^2 = sin(2x)

1-(cos x - sin x)²= sin(2x)

starting with the left side

1-(cosx -sinx)²

1-(cos²x + sin²x - 2sinx cos x)

We have to use following identities

cos²x + sin²x = 1 and 2 sinx cosx=sin2x

And on using these identities we get

1-(1-sin2x) = 1-1+sin2x=sin2x which is equal to the right side