Modulus of tangent of complex number

Question

I need to find real, imaginary parts of $\tan(x+yi)$ and the modulus of it. I have: $$\operatorname{Re}(\tan(x+yi))={\frac{\sin2x}{\cos2x+\cosh2x}}$$ and $$\operatorname{Im}(\tan(x+yi))={\frac{\sinh2y}{\cos2x+\cosh2x}}$$

I know that $|Z|={\sqrt{\operatorname{Re}^2+\operatorname{Im}^2}}$. But when I calculate with the results I've got, I don't get the actual answer on the book, which is $${\sqrt{{\frac{\cosh2y-\cos2x}{\cosh2y+\cos2x}}}}$$

Answer 1

You can observe that \begin{align} \sin^22x+\sinh^22y &=1-\cos^22x+\cosh^22y-1\\ &=\cosh^22y-\cos^22x\\ &=(\cosh2y+\cos2x)(\cosh2y-\cos2x) \end{align}