How did you go from $\omega RC + j$ to $\tan^{-1}\left( \frac{1}{\omega RC}\right)$ ?

You can write any complex number $\hat{z}=A+jB$ as a vector in the complex plane, where $A$ is the real axis component and $B$ is the imaginary axis component (here, in electronics world, $j=\sqrt{-1}$). The angle the vector makes with the real axis, by trigonometry, is $\theta = \tan^{-1}(B/A)$. Here, $\hat{z}= \omega RC + j$, and $A$, the real part is $\omega RC$; the imaginary part is $B=1$. Hence, $\theta$, the phase angle, is $\theta = \tan^{-1}\left(\frac{1}{\omega RC}\right)$.