Answer
$\frac{\sqrt 6+\sqrt 2}{4}$.
Work Step by Step
Rewrite the expression for sine of $105{}^\circ $ as the sum of two angles as,
$\sin 105{}^\circ =\sin \left( 60{}^\circ +45{}^\circ \right)$
Use the sum formula of sines and evaluate the modified expression as,
$\sin \left( 60{}^\circ +45{}^\circ \right)=\sin 60{}^\circ \cos 45{}^\circ +\cos 60{}^\circ \sin 45{}^\circ $
Substitute the values $\cos 45{}^\circ =\frac{1}{\sqrt{2}},\text{ }\cos 60{}^\circ =\frac{1}{2},\text{ }\sin 45{}^\circ =\frac{1}{\sqrt{2}},\text{ and }\sin 30{}^\circ =\frac{\sqrt{3}}{2}$.
$\begin{align}
& \sin \left( 60{}^\circ +45{}^\circ \right)=\left( \frac{\sqrt{3}}{2}\times \frac{1}{\sqrt{2}} \right)+\left( \frac{1}{2}\times \frac{1}{\sqrt{2}} \right) \\
& =\frac{\sqrt{3}}{2\sqrt{2}}+\frac{1}{2\sqrt{2}} \\
& =\frac{\sqrt{3}+1}{2\sqrt{2}}\\
&=\frac{\sqrt 6+\sqrt 2}{4}
\end{align}$
Hence, the exact value of $\sin 105{}^\circ $ is equivalent to $\frac{\sqrt 6+\sqrt 2}{4}$.
