\(tan\left(\frac{\pi}{3}-a\right)tan\left(\frac{\pi}{3}+a\right)=\frac{sin\left(\frac{\pi}{3}-a\right)sin\left(\frac{\pi}{3}+a\right)}{cos\left(\frac{\pi}{3}-a\right)cos\left(\frac{\pi}{3}+a\right)}\)
\(=\frac{cos2a-cos\frac{2\pi}{3}}{cos2a+cos\frac{2\pi}{3}}=\frac{cos2a+\frac{1}{2}}{cos2a-\frac{1}{2}}=\frac{2cos2a+1}{2cos2a-1}\)
\(\Rightarrow tana.tan\left(\frac{\pi}{3}-a\right)tan\left(\frac{\pi}{3}+a\right)=\frac{sina\left(2cos2a+1\right)}{cosa\left(2cos2a-1\right)}=\frac{2sina.cos2a+sina}{2cos2a.cosa-cosa}\)
\(=\frac{sin3a-sina+sina}{cos3a+cosa-cosa}=\frac{sin3a}{cos3a}=tan3a\)