Chapter 6 - Analytic Trigonometry - Section 6.7 Product-to-Sum and Sum-to-Product Formulas - 6.7 Assess Your Understanding - Page 524: 36

See below.

Use the Sum-to-Product Formula, we have: $LHS=\frac{cos(4\theta)-cos(8\theta)}{cos(4\theta)+cos(8\theta)}=\frac{-2sin(\frac{4\theta+8\theta}{2})sin(\frac{4\theta-8\theta}{2})}{2cos(\frac{4\theta+8\theta}{2})cos(\frac{4\theta-8\theta}{2})} =\frac{sin(6\theta)sin(2\theta)}{cos(6\theta)cos(2\theta)}=tan(6\theta)tan(2\theta)=RHS$