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