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

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

\begin{gather*}
	\text{let} \quad \sec(\theta) = x \\
	x(e^{i\theta} + e^{-i\theta}) = 2 \\
	e^{i\theta} + e^{-i\theta} = \frac{2}{x} \\
	(e^{i\theta})^2 + 1 = \frac{2}{x}e^{i\theta} \\
	(e^{i\theta})^2 + (-\frac{2}{x})e^{i\theta} + 1 = 0
\end{gather*}

\begin{gather*}
	e^{i\theta} = \frac{-(-\frac{2}{x}) \pm \sqrt{(-\frac{2}{x})^2 - 4}}{2} = x^{-1} \pm \sqrt{x^{-2} - 1} \\
	i\theta = \ln(x^{-1} \pm \sqrt{x^{-2} - 1}) \\
	\theta = -i\ln(x^{-1} \pm \sqrt{x^{-2} - 1})
\end{gather*}

\begin{gather*}
	\therefore \text{arcsec}(\theta) = -i\ln(\theta^{-1} \pm \sqrt{\theta^{-2} - 1})
\end{gather*}
\end{document}
