Chapter 6 - Analytic Trigonometry - Section 6.7 Product-to-Sum and Sum-to-Product Formulas - 6.7 Assess Your Understanding - Page 524: 33

See below.

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