Answer
$\sin 45^\circ+\sin 15^\circ=\sin 75^\circ$
Work Step by Step
Start with the left side:
$\sin 45^\circ+\sin 15^\circ$
Use the Sum-to-Product Formulas on page 560:
$=2\sin\frac{45^\circ+15^\circ}{2}\cos\frac{45^\circ-15^\circ}{2}$
Simplify:
$=2\sin\frac{60^\circ}{2}\cos\frac{30^\circ}{2}$
$=2\sin 30^\circ\cos 15^\circ$
$=2*\frac{1}{2}\cos 15^\circ$
$=\cos 15^\circ$
Use the fact that $\sin(90^\circ-x)=\cos x$:
$=\sin(90^\circ-15^\circ)$
$=\sin 75^\circ$
Since this equals the right side, the identity has been proven.
