\documentclass{../style}
\usepackage{amsmath}
\usepackage{amssymb}
\begin{document}
\euler

\begin{gather*}
	\therefore \sin(\theta) = \frac{e^{i\theta} - e^{-i\theta}}{2i} \\
	\cos(\theta) = \frac{e^{i\theta} + e^{-i\theta}}{2} \\
	\tan(\theta) = \frac{\sin(\theta)}{\cos{\theta}} = -\frac{i(-1 + e^{2i\theta})}{1 + e^{2i\theta}} \\
	\cot(\theta) = \frac{1}{\tan(\theta)} = -\frac{1 + e^{2i\theta}}{i(-1 + e^{2i\theta})}
\end{gather*}

\begin{gather*}
	\text{let} \quad \cot(\theta) = x \\
	x(i + ie^{2i\theta}) = -1(1 + e^{2i\theta}) \\
	ix + ixe^{2i\theta} = -e^{2i\theta} - 1 \\
	ixe^{2i\theta} + e^{2i\theta} = -1 - ix \\
	(ix + 1)e^{2i\theta} = -1 - ix \\
	e^{2i\theta} = -\frac{1 - ix}{1 + ix} \\
	2i\theta = \ln(\frac{x + i}{x - i}) \\
	i\theta = \frac{1}{2}\ln(\frac{x + i}{x - i}) \\
	\theta = -\frac{i}{2}\ln(\frac{x + i}{x - i})
\end{gather*}

\begin{gather*}
	\therefore \text{arccot}(\theta) = -\frac{i}{2}\ln(\frac{\theta + i}{\theta - i})
\end{gather*}
\end{document}