Question

If cosθ + sinθ = √2cosθ
Prove that: cosθ - sinθ = √2sinθ

Answer 1

cosθ + sinθ = √2cosθ
squaring both sides we get : cos^2θ + sin^2θ+2cosθsinθ = 2cos^2(θ)
=> 1+ 2cosθsinθ = 2cos^2(θ) => 2cosθsinθ = 2cos^2(θ)-1

now cosθ - sinθ ( squaring ) we get : 1-2cosθsinθ = 1-(2cos^2(θ)-1) = 1+1- 2cos^2(θ)
= 2- 2cos^2(θ) = 2[(1-cos^2(θ)] = 2sin^2θ
=>(cosθ - sinθ)^2=2sin^2θ ( cancelling square on L.H.S) we get cosθ - sinθ = √2sinθ ( hence proved )

Answer 2

thanx