Question

Let $P=\{\theta :\sin \theta -\cos \theta =\sqrt{2}\cos \theta \}$ and $Q=\{\theta :\sin \theta +\cos \theta =\sqrt{2}\sin \theta \}$ be two sets. Then:

• A. $P\subset Q$ and $Q-P\ne \varphi$
• B. $Q\text{/}\subset P$
• C. $P=Q$
• D. $P\text{/}\subset Q$

Answer 1

The correct option is C $P=Q$

$\sin \theta -\cos \theta =\sqrt{2}\cos \theta$
$\Rightarrow \sin \theta =\cos \theta +\sqrt{2}\cos \theta$
$=(\sqrt{2}+1)\cos \theta =\left(\frac{2-1}{\sqrt{2}-1}\right)\cos \theta$
$\Rightarrow (\sqrt{2}-1)\sin \theta =\cos \theta$
$\Rightarrow \sin \theta +\cos \theta =\sqrt{2}\sin \theta$
$\therefore P=Q$