Chapter 7 - Section 7.3 - Double-Angle, Half-Angle, and Product-Sum Formulas - 7.3 Exercises - Page 561: 67

$\frac{\sqrt{2}+\sqrt{3}}{2}$

Use the fourth Product-to-Sum Formula, $\sin u\sin v=\frac{1}{2}[\cos (u-v)-\cos(u+v)]$. $2\sin 52.5^\circ\sin97.5^\circ$ $=2\times\frac{1}{2}[\cos(52.5^\circ-97.5^\circ)-\cos (52.5^\circ+97.5^\circ)]$ $=\cos(-45^\circ)-\cos(150^\circ)$ $=\frac{\sqrt{2}}{2}-(-\frac{\sqrt{3}}{2})$ $=\frac{\sqrt{2}+\sqrt{3}}{2}$