Question

If cos theta -sin theta =2underoot sin theta prove that cos theta +sintheta =underoot 2 costheta

Answer 1

Given that

cosθ - sinθ = (√2 )sinθ

⇒ cosθ = sinθ +  (√2 )sinθ

⇒ cosθ = sinθ (1+√2 )

Multiplying both side with (1-√2 ), we get

⇒ (1-√2 )cosθ = sinθ (1+√2 )(1-√2 )

⇒ cosθ - (√2 )cosθ = sinθ (1 - 2)               [ using a²-b² = (a+b)(a-b) ]

⇒ cosθ - (√2 )cosθ = - sinθ

⇒ cosθ + sinθ = (√2 )cosθ

Hence Proved.