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