Answer
$\frac{\sqrt{2}-\sqrt{6}}{4}$
Work Step by Step
$\cos(105^\circ)$
$=\cos(60^\circ+45^\circ)$
Use the addition formula for cosine on page 545:
$=\cos(60^\circ)\cos(45^\circ)-\sin(60^\circ)\sin(45^\circ)$
$=\frac{1}{2}*\frac{\sqrt{2}}{2}-\frac{\sqrt{3}}{2}*\frac{\sqrt{2}}{2}$
$=\frac{\sqrt{2}}{4}-\frac{\sqrt{6}}{4}$
$=\frac{\sqrt{2}-\sqrt{6}}{4}$
