 Algebra of complex numbers
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# Let  and $\mathrm{i}=\sqrt{-1}$ then  $\mathrm{r}=\sqrt{\left({\mathrm{a}}^{2}+{\mathrm{b}}^{2}\right)}=|\mathrm{z}|$and  $\mathrm{\theta }={\mathrm{tan}}^{-1}\left(\frac{\mathrm{b}}{\mathrm{a}}\right)=\mathrm{arg}\left(\mathrm{z}\right)$now, $|\mathrm{z}{|}^{2}={\mathrm{a}}^{2}+{\mathrm{b}}^{2}=\left(\mathrm{a}+\mathrm{ib}\right)\left(\mathrm{a}-\mathrm{ib}\right)=\mathrm{z}\overline{\mathrm{z}}$and  $\left|{\mathrm{z}}_{1}{\mathrm{z}}_{2}{\mathrm{z}}_{3}\dots .{\mathrm{z}}_{\mathrm{n}}\right|=\left|{\mathrm{z}}_{1}\right|\left|{\mathrm{z}}_{2}\right|\left|{\mathrm{z}}_{3}\right|\dots \left|{\mathrm{z}}_{\mathrm{n}}\right|$if  then f(z) is called unimodular. In this casef(z) can always be expressed as $\mathrm{f}\left(\mathrm{z}\right)={\mathrm{e}}^{\mathrm{i\alpha }},\mathrm{\alpha }\in \mathrm{R}$Also, ${\mathrm{e}}^{\mathrm{i\alpha }}+{\mathrm{e}}^{\mathrm{i\beta }}={\mathrm{e}}^{\mathrm{i}\left(\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\right)}2\mathrm{cos}\left(\frac{\mathrm{\alpha }-\mathrm{\beta }}{2}\right)$  and ${\mathrm{e}}^{\mathrm{i\alpha }}-{\mathrm{e}}^{\mathrm{i\beta }}={\mathrm{e}}^{\mathrm{i}\left(\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\right)}2\mathrm{isin}\left(\frac{\mathrm{\alpha }-\mathrm{\beta }}{2}\right)$ where  $\mathrm{\alpha },\mathrm{\beta }\in \mathrm{R}$

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## now,$\begin{array}{l}\mathrm{cot}\left(\mathrm{arg}\mathrm{z}\right)-\frac{6}{\mathrm{z}}=\mathrm{cot}\left(\frac{\mathrm{\pi }}{4}+\frac{\mathrm{\alpha }}{2}\right)-\frac{6}{6{\mathrm{e}}^{\left(\frac{\mathrm{\pi }}{4}+\frac{\mathrm{\alpha }}{2}\right)}\mathrm{cos}\left(\frac{\mathrm{\pi }}{4}-\frac{\mathrm{\alpha }}{2}\right)}\\ =\mathrm{cot}\left(\frac{\mathrm{\pi }}{4}+\frac{\mathrm{\alpha }}{2}\right)-\frac{{\mathrm{e}}^{-1}\left(\frac{\mathrm{\pi }}{4}+\frac{\mathrm{\alpha }}{2}\right)}{\mathrm{cos}\left(\frac{\mathrm{\pi }}{2}-\left(\frac{\mathrm{\pi }}{4}+\frac{\mathrm{\alpha }}{2}\right)\right)}\\ =\mathrm{cot}\left(\frac{\mathrm{\pi }}{4}+\frac{\mathrm{\alpha }}{2}\right)-\frac{\left\{\mathrm{cos}\left(\frac{\mathrm{\pi }}{4}+\frac{\mathrm{\alpha }}{2}\right)-\mathrm{isin}\left(\frac{\mathrm{\pi }}{4}+\frac{\mathrm{\alpha }}{2}\right)\right\}}{\mathrm{sin}\left(\frac{\mathrm{\pi }}{4}+\frac{\mathrm{\alpha }}{2}\right)}}\\ =\mathrm{cot}\left(\frac{\mathrm{\pi }}{4}+\frac{\mathrm{\alpha }}{2}\right)-\mathrm{cot}\left(\frac{\mathrm{\pi }}{4}+\frac{\mathrm{\alpha }}{2}\right)+\mathrm{i}\\ =\mathrm{i}\end{array}$

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