Answer
$=2-\sqrt{3}$
Work Step by Step
$\tan(15^\circ)$
$=\tan(45^\circ-30^\circ)$
Use the subtraction formula for tangent on page 545:
$=\frac{\tan 45^\circ-\tan 30^\circ}{1+\tan 45^\circ \tan 30^\circ}$
Simplify:
$=\frac{1-\frac{\sqrt{3}}{3}}{1+1*\frac{\sqrt{3}}{3}}$
Multiply top and bottom by 3:
$=\frac{(1-\frac{\sqrt{3}}{3})*3}{(1+\frac{\sqrt{3}}{3})*3}$
$=\frac{3-\sqrt{3}}{3+\sqrt{3}}$
Multiply top and bottom by $3-\sqrt{3}$ and simplify:
$=\frac{(3-\sqrt{3})(3-\sqrt{3})}{(3+\sqrt{3})(3-\sqrt{3})}$
$=\frac{9-6\sqrt{3}+3}{9-3}$
$=\frac{12-6\sqrt{3}}{6}$
$=2-\sqrt{3}$
