Question

If $cosec \, \theta = \frac{p + q}{p - q} \left(p\ne q\ne0\right)$, then $\left|\cot \left(\frac{\pi}{4}+\frac{\theta}{2}\right)\right|$ is equal to :

• A. $\sqrt{\frac{p}{q}}$
• B. $\sqrt{\frac{q}{p}}$
• C. $\sqrt{pq}$
• D. $pq$

Answer 1

Answer: (B)

$\left|\cot\left(\frac{\pi}{4}+\frac{\theta}{2}\right)\right|=\left|\frac{1-\tan \frac{\theta}{2}}{1+\tan \frac{\theta}{2}}\right|$
$=\frac{\cos \frac{\theta}{2}-\sin \frac{\theta}{2}}{\cos \frac{\theta }{2}+\sin \frac{\theta}{2}}\times \frac{\cos \frac{\theta}{2}-\sin \frac{\theta}{2}}{\cos \frac{\theta}{2}-\sin \frac{\theta}{2}}$
$=\frac{\cos^{2} \frac{\theta }{2}+\sin^{2} \frac{\theta}{2}-2\sin \frac{\theta}{2} \cos\frac{\theta}{2}}{\cos\,\theta}=\frac{1-\sin\,\theta}{\cos\,\theta}$
$=\frac{1-\frac{p-q}{p+q}}{\sqrt{1-\left(\frac{p-q}{p+q}\right)^{2}}}=\frac{\sqrt{q}}{\sqrt{p}}$