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 $Q-P \neq \varnothing$

JEE Advanced · 2011 · 10 Apr 2011 · Paper 1 · Trigonometric Equations

Let

P = {θ : sin θ - cos θ = 2 cos θ}

and

Q = {θ : sin θ + cos θ = 2 sin θ}

be two sets. Then

Answer: (D)

For set

P

sin θ - cos θ = 2 cos θ

gives

sin θ = (2 + 1) cos θ

, so

tan θ = 2 + 1

.

For set

Q

sin θ + cos θ = 2 sin θ

gives

(2 - 1) sin θ = cos θ

, so

tan θ = \frac{1}{2 - 1} = \frac{2 + 1}{( 2 - 1 ) ( 2 + 1 )} = 2 + 1

.

Since both sets have the same condition

tan θ = 2 + 1

, we have

P = Q

JEE Advanced · 2011 · 10 Apr 2011 · Paper 1 · Single correct option