Question

If cos θ + sin θ = √2 cos θ, find cos θ - sin θ

Answer 1

Answer:

Wehave,

→cosθ +sin θ =√2cos θ .

[ Squaring both side, we get ] .

⇒(cosθ+sinθ)²=2cos²θ.

⇒cos²θ+sin²θ+2cosθsinθ=2cos².

⇒sin²θ+2cosθsinθ=2cos²θ-cos²θ.

⇒sin²θ+2cosθsinθ=cos²θ.

⇒cos²θ-2cosθsinθ=sin²θ.

[ Adding sin²θ both side, we get ] .

⇒cos²θ-2cosθsinθ+sin²θ=sin²θ+sin²θ.

⇒(cosθ-sinθ)²=2sin²θ.

⇒cosθ-sinθ=√(2sin²θ).

∴cosθ-sinθ=√2sinθ.

Hence,itis proved.