Chapter 7 - Analytic Trigonometry - 7.5 Sum and Difference Formulas - 7.5 Assess Your Understanding - Page 487: 32

$-1$

Recall that: $\cos(\alpha+\beta)=\cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta)$ Hence, $\cos(\frac{5\pi}{12})\cos(\frac{7\pi}{12})-\sin(\frac{5\pi}{12})\sin(\frac{7\pi}{12})=\cos\left(\frac{5\pi}{12}+\frac{7\pi}{12}\right)=\cos(\pi)=-1$