Using
R425: Euler's formulas for a real variable, we have
\begin{equation}
z
= r e^{i \theta}
= r \cos \theta + i r \sin \theta
= (r \cos \theta, r \sin \theta)
\end{equation}
Using the definition of complex conjugation and the Euler formula in the other direction, we thus have
\begin{equation}
\overline{z}
= \overline{(r \cos \theta, r \sin \theta)}
= (r \cos \theta, - r \sin \theta)
= r \cos \theta - i r \sin \theta
= r e^{- i \theta}
\end{equation}
$\square$
