$\tan \left( 105{}^\circ  \right)=\tan \left( 45{}^\circ +60{}^\circ  \right)=\frac{\tan 45{}^\circ +\tan 60{}^\circ }{1-\tan 45{}^\circ \tan 60{}^\circ }=\frac{1+\sqrt{3}}{1-1\times \sqrt{3}}=\frac{1+\sqrt{3}}{1-\sqrt{3}}\times \frac{1+\sqrt{3}}{1+\sqrt{3}}=-\frac{\left( \sqrt{3}+1 \right)2}{2}$