Answer
$$\frac{(n-ik)^2-1}{(n+ik)^2+2}=\frac{n^2-2ikn+i^2k^2-1}{n^2+2ikn+i^2k^2+2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{(n-ik)^2-1}{(n+ik)^2+2}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{n^2-2ikn+i^2k^2-1}{n^2+2ikn+i^2k^2+2}\end{aligned} $$ | |
| ① | Find $ \left(n-ik\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ n } $ and $ B = \color{red}{ ik }$. $$ \begin{aligned}\left(n-ik\right)^2 = \color{blue}{n^2} -2 \cdot n \cdot ik + \color{red}{\left( ik \right)^2} = n^2-2ikn+i^2k^2\end{aligned} $$ |
| ② | Find $ \left(n+ik\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ n } $ and $ B = \color{red}{ ik }$. $$ \begin{aligned}\left(n+ik\right)^2 = \color{blue}{n^2} +2 \cdot n \cdot ik + \color{red}{\left( ik \right)^2} = n^2+2ikn+i^2k^2\end{aligned} $$ |
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