Let z=x+iy. Verify that |sinhz|^2 = (sinhx)^2 + (siny)^2 and |coshz|^2 = (sinhx)^2 + (cosy)^2.
Any suggestions?
This is plug and chug. $\displaystyle \sinh(z)=\frac{e^{x+iy}-e^{-(x+iy)}}{2}$. Go from there and remember that $\displaystyle \left|z^2\right|=\left|z\right|^2$