Answer
$\frac{\sqrt 6+\sqrt 2}{4}$
Work Step by Step
$\sin105^{\circ}=\sin(60+45)$
use formula $\sin(A+B)$ = $\sin A \cos B + \cos A \sin B$
$\sin(60+45) = \sin 60 \cos 45 + \cos 60 \sin 45 $
$= \frac{\sqrt 3}{2}.\frac{1}{\sqrt 2}+\frac{1}{2}.\frac{1}{\sqrt 2}$
$= \frac{\sqrt 3}{2\sqrt 2}+\frac{1}{2\sqrt 2}$
$= \frac{\sqrt 3+1}{2\sqrt 2}$
$\sin105^{\circ}=\frac{\sqrt 3+1}{2\sqrt 2}.\frac{\sqrt 2}{\sqrt 2}$
$\sin105^{\circ}=\frac{\sqrt 6+\sqrt 2}{4}$
