Question

If cosθ + sinθ = 1, then prove that cosθ − sinθ = ±1.

Answer 1

Given: cosθ + sinθ = 1

On squaring both sides, we get

(cosθ + sinθ)² = (1)²

⇒ cos² θ + sin² θ + 2sinθ cos θ = 1

⇒ cos²θ + sin²θ = cos²θ + sin²θ – 2sinθ cosθ  ...[∵ cos²θ + sin²θ = 1]

⇒ cos²θ + sin²θ = (cosθ – sinθ)² ...[∵ (a – b)² = (a² + b² – 2ab)]

⇒ 1 = (cosθ – sinθ)²

⇒ (cosθ – sinθ) = ±1

Hence Proved.