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  4. Let P = ( θ ; sin θ - cosθ = √2 cos θ and Q = θ; sinθ + cos θ = √2 sin θ be two sets. Then ,

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Q. 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 ,

AMUAMU 2017

Solution:

For set $P$,
$\Rightarrow \sin \theta-\cos \theta=\sqrt{2} \cos \theta$
$\Rightarrow \tan \theta-1=\sqrt{2} $
[dividing both sides by $\cos \theta$ ]
$\Rightarrow \tan \theta=\sqrt{2}+1\,\,\, ...$(i)
For set $Q$
$\sin \theta+\cos \theta=\sqrt{2} \sin \theta$
$\Rightarrow 1+\cot \theta=\sqrt{2}$
[dividing both sides by $\sin \theta$ ]
$\Rightarrow \cot \theta=\sqrt{2}-1$
$\Rightarrow \tan \theta=\sqrt{2}+1\,\,\,...(ii)$
Hence, $P=Q$
[from Eqs. (i) and (ii)]