To Prove: (sin x – cos x)² = 1 – sin 2x

Taking LHS,

= (sin x – cos x)²

Using,

(a – b)² = (a² + b² – 2ab)

= sin²x + cos²x – 2sinx cosx

= (sin²x + cos²x) – 2sinx cosx

= 1 – 2sinx cosx [∵ cos² θ + sin² θ = 1]

= 1 – sin2x [∵ sin 2x = 2 sinx cosx]

= RHS

∴ LHS = RHS

Hence Proved