Question

If $cos\Theta +sin\Theta =\sqrt{2}cos\Theta$, then show that $cos\Theta -sin\Theta =\sqrt{2}sin\Theta$.

Answer 1

Given $\cos \theta +\sin \theta =\sqrt{2}\cos \theta$

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

$⟹\frac{1}{\sqrt{2}-1}\sin \theta =\cos \theta$

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

$⟹\cos \theta =\frac{\sqrt{2}+1}{2-1}\sin \theta$

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

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