Answer
The expression $\sin 25{}^\circ \cos 5{}^\circ +\cos 25{}^\circ \sin 5{}^\circ $ is written as $\sin 30{}^\circ $ and the exact value of $\sin 30{}^\circ $ is $\frac{1}{2}$.
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,
$\begin{align}
& \sin \left( 25{}^\circ +5{}^\circ \right)=\sin 25{}^\circ \cos 5{}^\circ +\cos 25{}^\circ \sin 5{}^\circ \\
& \sin \left( 30{}^\circ \right)=\sin 25{}^\circ \cos 5{}^\circ +\cos 25{}^\circ \sin 5{}^\circ
\end{align}$
Therefore, the expression $\sin 25{}^\circ \cos 5{}^\circ +\cos 25{}^\circ \sin 5{}^\circ $ is equivalent to $\sin 30{}^\circ $.
From the knowledge of trigonometric ratios defined for sine of an angle, the exact value of $\sin 30{}^\circ $ is $\frac{1}{2}$.
