Trigonometry (10th Edition)

$tan~15^{\circ} = 2-\sqrt{3}$
Note that triangle $ACD$ is a right triangle since angle $ACD = 90^{\circ}$ We can use triangle $ACD$ to find $tan~15^{\circ}$: $tan~15^{\circ} = \frac{opposite}{adjacent}$ $tan~15^{\circ} = \frac{AC}{CD}$ $tan~15^{\circ} = \frac{1}{2+\sqrt{3}}$ $tan~15^{\circ} = \frac{1}{2+\sqrt{3}}~\times \frac{2-\sqrt{3}}{2-\sqrt{3}}$ $tan~15^{\circ} = \frac{2-\sqrt{3}}{1}$ $tan~15^{\circ} = 2-\sqrt{3}$