Deriving $\sin{ix}=i\sinh{x}$

Multiply the denominator and the numerator by $i.$

Re-arrangement of the representation of your result:

$$\boxed{\begin{align} \sin{ix} = -\frac{\sinh{x}}{i} = (-1)\cdot \frac{\sinh{x}}{i} = (i^2)\cdot \frac{\sinh{x}}{i} = i\sinh x \end{align}}$$