Answer
$$\tan105^\circ=-2-\sqrt 3$$
F is the correct choice.
Work Step by Step
$$\tan105^\circ$$
$105^\circ$ can be rewritten as the sum of $45^\circ$ and $60^\circ$, which means $\tan105^\circ$ would become $$\tan(45^\circ+60^\circ)$$
Now, we can apply the identity of $\tan$ of a sum $$=\frac{\tan60^\circ+\tan45^\circ}{1-\tan60^\circ\tan45^\circ}$$ $$=\frac{\sqrt 3+1}{1-\sqrt 3\times1}$$ $$=\frac{1+\sqrt 3}{1-\sqrt 3}$$ $$=\frac{1+\sqrt3}{1-\sqrt 3}\times\frac{1+\sqrt 3}{1+\sqrt 3}$$ (to simplify it even further) $$=\frac{(1+\sqrt 3)^2}{(1-3)}$$ $$=\frac{4+2\sqrt 3}{-2}$$ $$=-2-\sqrt 3$$
Therefore, F is the correct choice.