Answer
$$\tan15^\circ=2-\sqrt 3$$
E is the correct choice.
Work Step by Step
$$\tan15^\circ$$
$15^\circ$ can be rewritten as the difference of $45^\circ$ and $30^\circ$, which means $\tan15^\circ$ would become $$\tan(45^\circ-30^\circ)$$
Now, we can apply the identity of $\tan$ of a difference $$=\frac{\tan45^\circ-\tan30^\circ}{1+\tan45^\circ\tan30^\circ}$$ $$=\frac{1-\frac{1}{\sqrt 3}}{1+1\times\frac{1}{\sqrt 3}}$$ $$=\frac{1-\frac{\sqrt 3}{3}}{1+\frac{\sqrt 3}{3}}$$ $$=\frac{\frac{3-\sqrt 3}{3}}{\frac{3+\sqrt 3}{3}}$$ $$=\frac{3-\sqrt 3}{3+\sqrt 3}$$ $$=\frac{3-\sqrt3}{3+\sqrt 3}\times\frac{3-\sqrt 3}{3-\sqrt 3}$$ (to simplify it even further) $$=\frac{(3-\sqrt 3)^2}{(9-3)}$$ $$=\frac{12-6\sqrt 3}{6}$$ $$=2-\sqrt 3$$
Therefore, E is the correct choice.