Let P = { : sin - cos = 2 cos } and Q = { : sin + cos = 2 sin } be two sets. Then [2011]

Question

Let $P = {θ : \sin θ - \cos θ = \sqrt{2} \cos θ}$ and $Q = {θ : \sin θ + \cos θ = \sqrt{2} \sin θ}$ be two sets. Then [2011]

Smart Achievers · Accepted Answer

(4)

$P = {θ : \sin θ - \cos θ = \sqrt{2} \cos θ}$

$\Rightarrow \sin θ = (\sqrt{2} + 1) \cos θ \Rightarrow \tan θ = \sqrt{2} + 1$

$Now, Q = {θ : \sin θ + \cos θ = \sqrt{2} \sin θ}$

$\Rightarrow \cos θ = (\sqrt{2} - 1) \sin θ \Rightarrow \tan θ = \frac{1}{\sqrt{2} - 1} \times \frac{\sqrt{2} + 1}{\sqrt{2} + 1}$

$\Rightarrow \tan θ = \sqrt{2} + 1$

$∴$ $P = Q$

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