Chapter 9 - Trigonometric Identities, Models, and Complex Numbers - 9.5 Sums with Different Periods and Acoustic Beats - Exercises and Problems for Section 9.5 - Exercises and Problems - Page 385: 1

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

Finding the exact value using the Laws of Trigonometry, we obtain: $$\left(-2\sin \left(\frac{165^{\circ \:}+75^{\circ \:}}{2}\right)\sin \left(\frac{165^{\circ \:}-75^{\circ \:}}{2}\right)\right) \\ -2\cdot \frac{\sqrt{3}}{2}\cdot \frac{\sqrt{2}}{2} \\ -\sqrt{\frac{3}{2}}$$