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 $Q-P\ne \varnothing$
• B. $Q\subset P$
• C. $P\subset Q$
• D. $P=Q$

Answer 1

Answer: (D)

In set $P,\sin \theta =(\sqrt{2}+1)\cos \theta$
$\Rightarrow \tan \theta =\sqrt{2}+1$
In set $Q,(\sqrt{2}-1)\sin \theta =\cos \theta$
$\Rightarrow \tan \theta =\frac{1}{\sqrt{2}-1}=\sqrt{2}+1$
$\Rightarrow P=Q$