Answer
$\frac{1+\sqrt3}{2\sqrt2}$
Work Step by Step
Using the given formula and the known values: $\sin105^\circ=\sin60^\circ\cos45^\circ+\sin45^\circ\cos60^\circ=\frac{\sqrt3}{2}\frac{1}{\sqrt2}+\frac{1}{\sqrt2}\frac{1}{2}=\frac{1+\sqrt3}{2\sqrt2}$
