Question

If $cos\theta-sin\theta=\sqrt{2}sin\theta$. Prove that $cos\theta+sin\theta=\sqrt{2}cos\theta$​

Answer 1

Step-by-step explanation:

Given Equation is cosθ - sinθ = √2 sinθ

It can be written as,

⇒ (cosθ - sinθ)² = 2sin²θ

⇒ cos²θ + sin²θ - 2cosθsinθ = sin²θ + sin²θ

⇒ sin²θ = cos²θ - 2sinθcosθ

⇒ cos²θ + sin²θ + 2sinθcosθ = 2cos²θ

⇒ (cosθ + sinθ)² = 2cos²θ

⇒ cosθ + sinθ = √2cosθ

Hope it helps!

Answer 2

cosθ-sinθ=√2sinθ

or, cosθ=√2sinθ+sinθ

or, cosθ=sinθ(√2+1)

or, cosθ=sinθ(√2+1)(√2-1)/(√2-1)

or, √2cosθ-cosθ=sinθ{(√2)²-(1)²}

or, √2cosθ-cosθ=sinθ(2-1)

or, -cosθ-sinθ=-√2cosθ

or, cosθ+sinθ=√2cosθ