Question

If $cosec\theta =\frac{p+q}{p-q}\left(p\ne q\ne 0\right)$, 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)

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