The two sides are not equal.

**Left Hand Side:**

**Factorization**

\[\frac{\sin{\left(\frac{\pi}{6} \tan{\left(\frac{\pi}{3} \right)} \right)}}{\tan{\left(\frac{\pi}{6} \cos{\left(\frac{\pi}{3} \right)} \right)}}\]

**Result**

\[\frac{\sin{\left(\frac{\sqrt{3} \pi}{6} \right)}}{2 - \sqrt{3}}\]

**Numerical**

\[2.93935254788098\]

**Simplification**

\[\sqrt{3} \sin{\left(\frac{\sqrt{3} \pi}{6} \right)} + 2 \sin{\left(\frac{\sqrt{3} \pi}{6} \right)}\]

So $LHS\neq RHS$