Eq. \ref{eq:5563180} is an initial equation. $$\theta_{\rm Brewster} + \theta_{\rm refracted} = 90^{\circ} \label{eq:5563180}$$ Subtract $$\theta_{\rm Brewster}$$ from both sides of Eq. \ref{eq:5563180}; yields Eq. \ref{eq:3893026}. $$\theta_{\rm refracted} = 90^{\circ} - \theta_{\rm Brewster} \label{eq:3893026}$$ Eq. \ref{eq:9932375} is an initial equation. $$n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 ) \label{eq:9932375}$$ Change variable $$\theta_2$$ to $$\theta_{\rm refracted}$$ and $$\theta_1$$ to $$\theta_{\rm Brewster}$$ in Eq. \ref{eq:9932375}; yields Eq. \ref{eq:4176694}. $$n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( \theta_{\rm refracted} ) \label{eq:4176694}$$ Substitute LHS of Eq. \ref{eq:3893026} into Eq. \ref{eq:4176694}; yields Eq. \ref{eq:4962698}. $$n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} ) \label{eq:4962698}$$ Eq. \ref{eq:3940135} is an identity. $$\sin( 90^{\circ} - x ) = \cos( x ) \label{eq:3940135}$$ Change variable $$\theta_{\rm Brewster}$$ to $$x$$ in Eq. \ref{eq:3940135}; yields Eq. \ref{eq:7426234}. $$\sin( 90^{\circ} - \theta_{\rm Brewster} ) = \cos( \theta_{\rm Brewster} ) \label{eq:7426234}$$ Substitute LHS of Eq. \ref{eq:7426234} into Eq. \ref{eq:4962698}; yields Eq. \ref{eq:9701820}. $$n_1 \sin( \theta_{\rm Brewster} ) = n_2 \cos( \theta_{\rm Brewster} ) \label{eq:9701820}$$ Divide both sides of Eq. \ref{eq:9701820} by $$n_1$$; yields Eq. \ref{eq:9314305}. $$\sin( \theta_{\rm Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{\rm Brewster} ) \label{eq:9314305}$$ Divide both sides of Eq. \ref{eq:9314305} by $$\cos( \theta_{\rm Brewster} )$$; yields Eq. \ref{eq:8585856}. $$\frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} = \frac{n_2}{n_1} \label{eq:8585856}$$ Eq. \ref{eq:2621708} is an identity. $$\tan( x ) = \frac{ \sin( x )}{\cos( x )} \label{eq:2621708}$$ Change variable $$\theta_{\rm Brewster}$$ to $$x$$ in Eq. \ref{eq:2621708}; yields Eq. \ref{eq:1898054}. $$\tan( \theta_{\rm Brewster} ) = \frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} \label{eq:1898054}$$ Substitute LHS of Eq. \ref{eq:1898054} into Eq. \ref{eq:8585856}; yields Eq. \ref{eq:5179630}. $$\tan( \theta_{\rm Brewster} ) = \frac{ n_2 }{ n_1 } \label{eq:5179630}$$ Apply function $$\arctan{ x }$$ with argument $$x$$ to Eq. \ref{eq:5179630}; yields Eq. \ref{eq:8186016} $$\theta_{\rm Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) } \label{eq:8186016}$$ Eq. \ref{eq:8186016} is one of the final equations.