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 $Q\sim P\ne \varphi$
• B. $Q\not\subset P$
• C. $P\not\subset Q$
• D. $P=Q$.

Answer 1

Answer: (D)

Solution: $\sin \theta -\cos \theta =\sqrt{2}\cos \theta$
$\iff \sin \theta =(\sqrt{2}+1)\cos \theta$
$\iff \cos \theta =(\sqrt{2}-1)\sin \theta$
$\iff \sin \theta +\cos \theta =\sqrt{2}\sin \theta$
$\Rightarrow P=Q.$