\begin{align*} \left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right) & =a^{2}c^{2}+b^{2}d^{2}+a^{2}d^{2}+b^{2}c^{2}\\ & =\left(ac\pm bd\right)^{2}\mp2abcd+a^{2}d^{2}+b^{2}c^{2}\\ & =\left(ac\pm bd\right)^{2}+\left(ad\mp bc\right)^{2} \end{align*}

(0)-2

\begin{align*} \left\{ \Re^{2}\left(\alpha\right)+\Im^{2}\left(\alpha\right)\right\} \left\{ \Re^{2}\left(\beta\right)+\Im^{2}\left(\beta\right)\right\} & =\left|\alpha\right|^{2}\left|\beta\right|^{2}\\ & =\left|\alpha\beta\right|^{2}\\ & =\left|\left\{ \Re\left(\alpha\right)+\Im\left(\alpha\right)i\right\} \left\{ \Re\left(\beta\right)+\Im\left(\beta\right)\right\} i\right|^{2}\\ & =\left|\Re\left(\alpha\right)\Re\left(\beta\right)-\Im\left(\alpha\right)\Im\left(\beta\right)+i\left\{ \Re\left(\alpha\right)\Im\left(\beta\right)+\Im\left(\alpha\right)\Re\left(\beta\right)\right\} \right|^{2}\\ & =\left\{ \Re\left(\alpha\right)\Re\left(\beta\right)-\Im\left(\alpha\right)\Im\left(\beta\right)\right\} ^{2}+\left\{ \Re\left(\alpha\right)\Im\left(\beta\right)+\Im\left(\alpha\right)\Re\left(\beta\right)\right\} ^{2} \end{align*} ここで\(\Re\left(\alpha\right)=a\;,\;\Im\left(\alpha\right)=b\;,\;\Re\left(\beta\right)=c\;,\;\Im\left(\beta\right)=d\)とおくと、
\[ \left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)=\left(ac-bd\right)^{2}+\left(ad+bc\right)^{2} \] \(b\rightarrow-b\)と置き換えると、
\[ \left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)=\left(ac+bd\right)^{2}+\left(ad-bc\right)^{2} \] １つにまとめて、
\[ \left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)=\left(ac\pm bd\right)^{2}+\left(ad\mp bc\right)^{2} \] となる。