If cosθ + sinθ = √2 cosθ, prove that : cosθ – sinθ = √2sin�

1.	If cosθ + sinθ = √2 cosθ, prove that : cosθ – sinθ = √2sinθ.
Answer» We have, (cosθ + sinθ) = √2cosθ By squaring both sides, (cosθ + sinθ)² = 2cos²θ ⇒ cos²θ + sin²θ + 2sinθcosθ = 2 cos²θ (∵ (a + b)² = a² + b² + 2ab) ⇒ sin²θ = cos²θ − 2sinθcosθ ⇒ 2sin²θ = cos²θ − 2sinθcosθ + sin²θ ⇒ (cosθ − sinθ)² = 2sin²θ ⇒ cosθ − sinθ = √2sinθ (By taking squaring root of both sides) Hence Proved.

If cosθ + sinθ = √2 cosθ, prove that : cosθ – sinθ = √2sinθ.

Answer»

We have,

(cosθ + sinθ) = √2cosθ

By squaring both sides,

(cosθ + sinθ)² = 2cos²θ

⇒ cos²θ + sin²θ + 2sinθcosθ = 2 cos²θ

(∵ (a + b)² = a² + b² + 2ab)

⇒ sin²θ = cos²θ − 2sinθcosθ

⇒ 2sin²θ = cos²θ − 2sinθcosθ + sin²θ

⇒ (cosθ − sinθ)² = 2sin²θ

⇒ cosθ − sinθ = √2sinθ

(By taking squaring root of both sides)

Hence Proved.

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If cosθ + sinθ = √2 cosθ, prove that : cosθ – sinθ = √2sinθ.

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