If $\cos ec\theta =\frac{p+q}{p-q} then cot\left(\frac{\pi }{4}+\frac{\theta }{2}\right)=$

$\begin{array}{rcl}\mathrm{cosec\theta }& =& \frac{\mathrm{p}+\mathrm{q}}{\mathrm{p}-\mathrm{q}}\\ \frac{1}{\mathrm{sin\theta }}& =& \frac{\mathrm{p}+\mathrm{q}}{\mathrm{p}-\mathrm{q}}\end{array}$

by componendo and dividendo

$\begin{array}{rcl}\frac{1+\mathrm{sin\theta }}{1-\mathrm{sin\theta }}& =& \frac{2\mathrm{p}}{2\mathrm{q}}\\ \frac{1-\cos \left({\displaystyle \frac{\mathrm{\pi }}{2}}+\mathrm{\theta }\right)}{1+\cos \left({\displaystyle \frac{\mathrm{\pi }}{2}}+\mathrm{\theta }\right)}& =& \frac{\mathrm{p}}{\mathrm{q}}\\ \frac{2{\sin }^2\left(\frac{\mathrm{\pi }}{4}+{\displaystyle \frac{\mathrm{\theta }}{2}}\right)}{2{\cos }^2\left(\frac{\mathrm{\pi }}{4}+\frac{\mathrm{\theta }}{2}\right)}& =& \frac{\mathrm{p}}{\mathrm{q}}\\ {\tan }^2\left(\frac{\mathrm{\pi }}{4}+\frac{\mathrm{\theta }}{2}\right)& =& \frac{\mathrm{p}}{\mathrm{q}}\\ \cot \left(\frac{\mathrm{\pi }}{4}+\frac{\mathrm{\theta }}{2}\right)& =& \sqrt{\frac{\mathrm{q}}{\mathrm{p}}}\end{array}$