Answer
$\sqrt{3}-2$
Work Step by Step
$\tan(165^\circ)$
$=\tan(135^\circ+30^\circ)$
Use the addition formula for tangent on page 545:
$=\frac{\tan 135^\circ+\tan 30^\circ}{1-\tan 135^\circ \tan 30^\circ}$
Simplify:
$=\frac{-1+\frac{\sqrt{3}}{3}}{1-(-1)*\frac{\sqrt{3}}{3}}$
$=\frac{-1+\frac{\sqrt{3}}{3}}{1+\frac{\sqrt{3}}{3}}$
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:
$=\frac{(-3+\sqrt{3})(3-\sqrt{3})}{(3+\sqrt{3})(3-\sqrt{3})}$
$=\frac{-9+6\sqrt{3}-3}{9-3}$
$=\frac{-12+6\sqrt{3}}{6}$
$=-2+\sqrt{3}$
$=\sqrt{3}-2$
