$\begin{array}{l}
3\cos \alpha - \sin \alpha = 1 \Rightarrow \sin \alpha = 3\cos \alpha - 1\\
{\sin ^2}\alpha + {\cos ^2}\alpha = 1 \Rightarrow {\left( {3\cos \alpha - 1} \right)^2} + {\cos ^2}\alpha = 1\\
\Leftrightarrow 9{\cos ^2}\alpha - 6\cos \alpha + 1 + {\cos ^2}\alpha - 1 = 0\\
\Leftrightarrow 10{\cos ^2}\alpha - 6\cos \alpha = 0\\
\Leftrightarrow 2\cos \alpha \left( {5\cos \alpha - 3} \right) = 0\\
\Leftrightarrow 5\cos \alpha - 3 = 0\left( {vi\,{0^0} < \alpha < {{90}^0}\,nen\,\cos \alpha > 0} \right)\\
\Leftrightarrow \cos \alpha = \frac{3}{5} \Rightarrow \sin \alpha = 3\cos \alpha - 1 = 3.\frac{3}{5} - 1 = \frac{4}{5}\\
\Rightarrow \tan \alpha = \frac{{\sin \alpha }}{{\cos \alpha }} = \frac{4}{5}.\frac{5}{3} = \frac{4}{3}
\end{array}$
