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

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

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

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

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