Answer
$\frac{\sqrt 2-\sqrt 6}{4}$
Work Step by Step
Given $\cos105^{\circ}$= $\cos(60^{\circ}+45^{\circ})$
Use formula $\cos(a+b)$ = $ \cos A \cos B - \sin A \sin B $
$\cos(60^{\circ}+45^{\circ})$= $\cos60 \cos45 - \sin60 \sin45$
$\cos(60^{\circ}+45^{\circ})$= $\frac{1}{2}.\frac{1}{\sqrt 2}-\frac{\sqrt 3}{2}.\frac{1}{\sqrt 2}$
=$\frac{1}{2\sqrt 2}-\frac{\sqrt 3}{2\sqrt 2}$
=$\frac{1-\sqrt 3}{2\sqrt 2} $
$\cos105^{\circ}$=$\frac{1-\sqrt 3}{2\sqrt 2} \frac{\sqrt 2}{\sqrt 2}$
$\cos105^{\circ}$=$\frac{\sqrt 2-\sqrt 6}{4}$
