Question

If $cos\theta +sin\theta =\sqrt{2}cos\theta$
Then Prove that

$cos\theta -sin\theta =\sqrt{2}sin\theta$

Answer 1

$\cos \theta +\sin \theta =\sqrt{2}\cos \theta \Rightarrow \sin \theta =\cos \theta (\sqrt{2}-1)$

Multiplying $\sqrt{2}+1$ both sides

$\sin \theta (\sqrt{2}+1)=\cos \theta (2-1)\Rightarrow \sqrt{2}\sin \theta +\sin \theta =\cos \theta \Rightarrow \sqrt{2}sin\theta =\cos \theta -\sin \theta$