Answer
$$\cos105^\circ=\frac{\sqrt2-\sqrt6}{4}$$
Work Step by Step
$$\cos105^{\circ}$$
As in the hint: $$105^\circ=60^\circ+45^\circ$$
That means $$\cos105^\circ=\cos(60^\circ+45^\circ)$$
Now we apply cosine of a sum:
$$\cos105^\circ=\cos60^\circ\cos45^\circ-\sin60^\circ\sin45^\circ$$
$$\cos105^\circ=\frac{1}{2}\frac{\sqrt2}{2}-\frac{\sqrt3}{2}\frac{\sqrt2}{2}$$
$$\cos105^\circ=\frac{\sqrt2}{4}-\frac{\sqrt6}{4}$$
$$\cos105^\circ=\frac{\sqrt2-\sqrt6}{4}$$
