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 $Q \sim P \neq \phi$
• B. $Q \not \subset P$
• C. $P \not \subset Q$
• D. $P=Q$.

Answer 1

Answer: (D)

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