Complex analysis... I can't write the function $f(z) =\tan (z)$ as a real part and imaginary part

Question

How to write the function $f(z) =\tan (z)$ as a real part and imaginary part?

I have reached the form : \begin{align} \tan (z) &= \frac{\sin(z)}{\cos(z)}\\ &= \frac{e^{iz} - e^{-iz}}{ i (e^{iz}+ e^{-iz})} \end{align}

Answer 1

If $x,y\in\mathbb R$, then\begin{align}\tan(x+yi)&=\frac{\tan(x)+\tan(yi)}{1-\tan(x)\tan(yi)}\\&=\frac{\tan(x)+i\tanh(y)}{1-i\tan(x)\tanh(y)}\\&=\frac{\bigl(\tan(x)+i\tanh(y)\bigr)\bigl(1+i\tan(x)\tanh(y)\bigr)}{1+\tan^2(x)\tanh^2(y)}.\end{align}Can you do the rest?