$\text{Since we have: } \ cosh(x) = cos(ix) \ \text{ and } \ sinh(x) = -i \cdot sin(ix) \\
\text{By substituting and rearranging, we have the following: }$
$
\begin{gather}
\begin{aligned}
f(x,y) &= cosh(x) \cdot sin(y) + ( - sinh(x) \cdot cos(y)) \cdot i + C \cdot i \\\\
&= cos(ix) \cdot sin(y) - sin(ix) \cdot cos(y) + C \cdot i \\\\
&= sin(y - ix) + C \cdot i \\\\
&= sin(-i \cdot (x+iy)) + C \cdot i \\\\
&= - sin(i(x+iy)) + C \cdot i \\\\
&= -i \cdot sinh(x + iy) + C \cdot i \\\\
&= -i \cdot sinh(z) + C \cdot i \\\\
\end{aligned}
\end{gather}
$
$\text{I think the answer has a typo, I got} \ -i \cdot sinh(z) + C \cdot i \ \text{ instead of } cosh(z) + C \cdot i$
