Answer
$$\frac{\sqrt{2}-\sqrt{6}}{4}$$
Work Step by Step
Since
$$\cos(x+y)= \cos x\cos y- \sin x\sin y $$
Then
\begin{align*}
\cos(105^{\circ})&= \cos(45^{\circ}+60^{\circ})\\
&=\cos 45^{\circ}\cos 60^{\circ}- \sin45^{\circ}\sin 60^{\circ}\\
&= \frac{\sqrt{2}}{4}\sin \theta - \frac{\sqrt{2}\sqrt{3}}{4}\cos \theta\\
&= \frac{\sqrt{2}-\sqrt{6}}{4}
\end{align*}
