\documentclass[a4paper]{article}
\begin{document}
\begin{eqnarray*}
f (z)
 & = & \csc z \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
f (z)
 & = & \csc z \\
& = & \frac{1}{\sin z} \\
& = & \frac{1}{\sin (x + iy)} \\
& = & \frac{1}{\sin x \cos (iy) + \cos x \sin (iy)} \\
& = & \frac{1}{\sin x \cosh y + i \cos x \sinh y} \\
& = & \frac{1}{\sin x \cosh y + i \cos x \sinh y} \frac{\sin x \cosh y - i \cos x \sinh y}{\sin x \cosh y - i \cos x \sinh y} \\
& = & \frac{\sin x \cosh y - i \cos x \sinh y}{(\sin x \cosh y)^{2} + (\cos x \sinh y)^{2}} \\
& = & \underbrace{\frac{\sin x \cosh y}{(\sin x \cosh y)^{2} + (\cos x \sinh y)^{2}}}_{u}
 + i \underbrace{\frac{- \cos x \sinh y}{(\sin x \cosh y)^{2} + (\cos x \sinh y)^{2}}}_{v} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\partial u}{\partial x} - \frac{\partial v}{\partial y}
 & = & 0 \quad \wedge \quad \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} \, \, \, = \, \, \, 0 \\
\end{eqnarray*}
\end{document}
