<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,12pt]{article}
\newcommand{\ds}{\displaystyle}
\newcommand{\pl}{\partial}
\parindent=0pt
\begin{document}


{\bf Question}

Simplify $$\left( \frac{a+ib}{a-ib}\right)^2 -
\left(\frac{a-ib}{a+ib}\right)^2$$

\vspace{.25in}

{\bf Answer}

$\ds \left( \frac{a+ib}{a-ib}\right)^2 -
\left(\frac{a-ib}{a+ib}\right)^2$
\begin{eqnarray*} &amp; = &amp; \frac{(a+ib)^4 -
(a-ib)^4}{(a+ib)^2(a+ib)^2} \\ &amp; = &amp; \frac{(a^2+2aib - b^2)^2 -
(a^2-2aib-b^2)}{(a^2+b^2)^2}\\ &amp; = &amp; \frac{a^4 + 4a^3ib - 6a^2b^2
- 4aib^3 +b^4 - (a^4 - 4a^3ib - 6a^2b^2 + 4aib^3
+b^4)}{(a^2+b^2)^2}  \\ &amp; = &amp; \frac{8a^3bi - 8ab^3i}{(a^2+b^2)^2}
\\ &amp; = &amp; \frac{8iab(a^2-b^2) }{(a^2+b^2)^2}
\end{eqnarray*}


\end{document}
</pre></body></html>