If P = \θ : sinθ - cosθ = √(2)cosθ\ and Q = \θ : sinθ + cosθ = √(2)sinθ\ be two sets. Then,

If P = \θ : sinθ - cosθ = √(2)cosθ\ and Q = \θ : sinθ +… — JEE Main 2011 Mathematics PYQ

AIEEE 2011

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

Show answer and solution

Correct option: D

Solution

From set P: sinθ - cosθ = √2 cosθ → sinθ = (1+√2)cosθ → tanθ = 1+√2.

From set Q: sinθ + cosθ = √2 sinθ → cosθ = (√2-1)sinθ → tanθ = 1/(√2-1) = (√2+1)/1 = 1+√2 (after rationalizing).

Both sets give exactly tanθ = 1+√2, so P = Q.

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JEE Main 2011 Mathematics question on Trigonometry