Using only the trig ratios of 45 and 60 degrees, we use angle-sum to compute [tex]\tan 105^\circ=\tan(60^\circ+45^\circ)=\frac{\tan 60^\circ+\tan 45^\circ}{1-\tan 60^\circ\tan 45^\circ}=\frac{\sqrt{3}+1}{1-\sqrt{3}}=-2-\sqrt{3},[/tex] so [tex]\cot 105^\circ=\frac{1}{\tan 105^\circ}=-2+\sqrt{3}[/tex] and [tex]\tan 105^\circ-\cot105^\circ=\boxed{-2\sqrt{3}}.[/tex]
