If $cosec θ = \frac{p + q}{p - q},$ then $\cot (π / 4 + θ / 2) =$

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$\sqrt{\frac{p}{q}}$

$\sqrt{\frac{q}{p}}$

$\sqrt{p q}$

$p q$

answer is B.

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Detailed Solution

We have, $cosec θ = \frac{p + q}{p - q}$

Now,
$\cot (\frac{π}{4} + \frac{θ}{2}) = \frac{1 - \tan \frac{θ}{2}}{1 + \tan \frac{θ}{2}} = \frac{\cos \frac{θ}{2} - \sin \frac{θ}{2}}{\cos \frac{θ}{2} + \sin \frac{θ}{2}}$

$\Rightarrow \cot (\frac{π}{4} + \frac{θ}{2}) = \sqrt{{\frac{\cos \frac{0}{2} - \sin \frac{θ}{2}}{\cos \frac{θ}{2} + \sin \frac{θ}{2}})}^{2}} = \sqrt{\frac{1 - \sin θ}{1 + \sin θ}}$

$\Rightarrow c o t (\frac{π}{4} + \frac{θ}{2}) = \sqrt{\frac{\cos e c θ - 1}{\cos e c θ + 1}} = \sqrt{\frac{\frac{p + q}{p - q} - 1}{\frac{p + q}{p - q} + 1}} = \sqrt{\frac{q}{p}}$