Show that $\mathfrak{Re}(\textrm{Li}_2(e^{ix}))=\frac{x^2}{4}-\frac{\pi x}{2}+\frac{\pi^2}{6}$ (polylogarithm)

Question

I am working with the polylogarithm function and want to find closed expressions for $\textrm{Li}_2(e^{ix})$.

1. If I plot the function $\mathfrak{Re}(\textrm{Li}_2(e^{ix}))$ I get $y=\dfrac{x^2}{4}-\dfrac{\pi x}{2}+\dfrac{\pi^2}{6}$ in the interval from 0 to $2\pi$. How can I see this connection?

2. Is there something similar possible for $\mathfrak{Im}(\textrm{Li}_2(e^{ix}))$?

Edit: I saw this but was not able to use it.