Alternative Way to Prove $\tan^2 x+\tan^2 (x+60^{\circ})+\tan^2 (60^{\circ}-x)=9\tan^2 3x+6$

So in one of my recent post, I mentioned of one rare but extremely useful trigonometry identity that sounds:

$\tan^2 x+\tan^2 (x+60^{\circ})+\tan^2 (60^{\circ}-x)=9\tan^2 3x+6$

We should then use it when appropriate.