$\mathrm{lf} \mathrm{p}=\cos 2\mathrm{\alpha }+\mathrm{i} \sin 2\mathrm{\alpha }, \mathrm{q}=\cos 2\mathrm{\beta }+\mathrm{i} \sin 2\mathrm{\beta } \mathrm{then} \sqrt{\mathrm{p}/\mathrm{q}}-\sqrt{\mathrm{q}/\mathrm{p}} =$

$\begin{array}{rcl}\mathrm{p}& =& \cos 2\mathrm{\alpha }+\mathrm{i} \sin \mathrm{n}2\mathrm{\alpha }\end{array}$

$\begin{array}{rcl}\mathrm{p}& =& {\left(\cos \mathrm{\alpha }+\mathrm{i} \sin \mathrm{\alpha }\right)}^2\end{array}$

$\begin{array}{rcl} & \Rightarrow & \sqrt{\mathrm{p}}=\cos \mathrm{\alpha }+\mathrm{i} \sin \mathrm{\alpha }\end{array}$

$\begin{array}{rcl}\mathrm{q}& =& \cos 2\mathrm{\beta }+\mathrm{i} \sin 2\mathrm{\beta }={\left(\cos \mathrm{\beta }+\mathrm{i} \sin \mathrm{\beta }\right)}^2\end{array}$

$\begin{array}{rcl}\sqrt{\mathrm{q}}& =& \cos \mathrm{\beta }+\mathrm{i} \sin \mathrm{\beta }\end{array}$

$\begin{array}{rcl}\sqrt{\frac{\mathrm{p}}{\mathrm{q}}}& =& \frac{\cos \mathrm{\alpha }+\mathrm{i} \sin \mathrm{\alpha }}{\cos \mathrm{\beta }+\mathrm{i} \sin \mathrm{\beta }}=\frac{\mathrm{cis} \mathrm{\alpha }}{\mathrm{cis} \mathrm{\beta }}=\mathrm{cis}(\mathrm{\alpha }-\mathrm{\beta })\end{array}$

$\begin{array}{rcl}\sqrt{\frac{\mathrm{p}}{\mathrm{q}}}& =& \cos (\mathrm{\alpha }-\mathrm{\beta })+\mathrm{i} \sin (\mathrm{\alpha }-\mathrm{\beta })\end{array}$

$\begin{array}{rcl}\sqrt{\frac{\mathrm{q}}{\mathrm{p}}}& =& \cos (\mathrm{\alpha }-\mathrm{\beta })-\mathrm{i} \sin (\mathrm{\alpha }-\mathrm{\beta })\end{array}$

$\begin{array}{rcl} & & \sqrt{\frac{\mathrm{p}}{\mathrm{q}}}-\sqrt{\frac{\mathrm{q}}{\mathrm{p}}}\left(\cos \right(\mathrm{\alpha }-\mathrm{\beta })+\mathrm{i} \sin (\mathrm{\alpha }-\mathrm{\beta }\left)\right)-\left(\cos \right(\mathrm{\alpha }-\mathrm{\beta })-\mathrm{i} \sin (\mathrm{\alpha }-\mathrm{\beta }\left)\right)\end{array}$

$\begin{array}{rcl} & =& 2 \mathrm{i} \sin (\mathrm{\alpha }-\mathrm{\beta })\end{array}$