[tex]\displaystyle\bf\\\frac{Sin(30^\circ+\alpha )-Cos(60^\circ+\alpha )}{Sin(30^\circ+\alpha )+Cos(60^\circ+\alpha )} =\frac{Cos\Big[90^\circ-(30^\circ+\alpha )\Big]-Cos(60^\circ+\alpha )}{Cos\Big[90^\circ-(30^\circ+\alpha )\Big]+Cos(60^\circ+\alpha) } =[/tex]

[tex]\displaystyle\bf\\=\frac{Cos(60^\circ-\alpha )-Cos(60^\circ+\alpha )}{Cos(60^\circ-\alpha )+Cos(60^\circ+\alpha )} =\\\\\\=\frac{-2Sin\dfrac{60^\circ-\alpha -60^\circ-\alpha }{2} Sin\dfrac{60^\circ-\alpha +60^\circ+\alpha }{2} }{2Cos\dfrac{60^\circ-\alpha -60^\circ-\alpha }{2}Cos\dfrac{60^\circ-\alpha +60^\circ+\alpha }{2} } =\\\\\\=\frac{-2Sin(-2\alpha )\cdot Sin120^\circ}{2Cos(-2\alpha )\cdot Cos120^\circ} =[/tex]

[tex]\displaystyle\bf\\=\frac{Sin2\alpha \cdot\frac{\sqrt{3} }{2} }{Cos2\alpha \cdot(-\frac{1}{2}) }=-\sqrt{3} tg2\alpha[/tex]