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