Answer
See explanations.
Work Step by Step
Step 1. Using the Sum-to- Product Formulas, we have
$LHS=\frac{2sin(\frac{2x+6x}{2})cos(\frac{2x-6x}{2})}{2cos(\frac{2x+6x}{2})sin(\frac{2x-6x}{2})}=\frac{sin(4x)cos(2x)}{coos(4x)sin(-2x)}=-\frac{sin(4x)cos(2x)}{coos(4x)sin(2x)}$
Step 2. We have
$RHS=-tan(4x)cot(2x)=-\frac{sin(4x)}{cos(4x)}\times\frac{cos(2x)}{sin(2x)}$
Step 3. Since $LHS=RHS$, we verified the identity.
