Answer
The expression $\sin 40{}^\circ \cos 20{}^\circ +\cos 40{}^\circ \sin 20{}^\circ $ is written as $\sin 60{}^\circ $ and the exact value of $\sin 60{}^\circ $ is $\frac{\sqrt{3}}{2}$.
Work Step by Step
Use the sum formula of sines and rewrite the expression as the sum of angles to obtain the sine of the angle as,
$\begin{align}
& \sin \left( 40{}^\circ +20{}^\circ \right)=\sin 40{}^\circ \cos 20{}^\circ +\cos 40{}^\circ \sin 20{}^\circ \\
& \sin \left( 60{}^\circ \right)=\sin 40{}^\circ \cos 20{}^\circ +\cos 40{}^\circ \sin 20{}^\circ
\end{align}$
Therefore, the expression $\sin 40{}^\circ \cos 20{}^\circ +\cos 40{}^\circ \sin 20{}^\circ $ is equivalent to $\sin 60{}^\circ $.
From the knowledge of trigonometric ratios defined for sine of an angle, the exact value of $\sin 60{}^\circ $ is $\frac{\sqrt{3}}{2}$.
