Answer
$\frac{\sqrt{6}}{2}$
Work Step by Step
Use the first Sum-to-Product Formula, $\sin x+\sin y=2\sin \frac{x+y}{2}\cos\frac{x-y}{2}$.
$\sin 75^\circ+\sin 15^\circ$
$=2\sin \frac{75^\circ+15^\circ}{2}\cos \frac{75^\circ-15^\circ}{2}$
$=2\sin \frac{90^\circ}{2}\cos\frac{60^\circ}{2}$
$=2\sin 45^\circ\cos 30^\circ$
$=2\times\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}$
$=\frac{\sqrt{6}}{2}$
