If $\tan \theta \tan \left(\frac{\pi }{3}+\theta \right)\tan \left(-\frac{\pi }{3}+\theta \right)=k\tan 3\theta ,$ then the value of $k,$ is 

We have,

  $\tan \theta \tan \left(\frac{\pi }{3}+\theta \right)\tan \left(-\frac{\pi }{3}+\theta \right)=k\tan 3\theta$

$\begin{array}{l}\Rightarrow \tan \theta \left(\frac{\tan \theta +\sqrt{3}}{1-\sqrt{3}\tan \theta }\right)\left(\frac{\tan \theta -\sqrt{3}}{1+\sqrt{3}\tan \theta }\right)=k\tan 3\theta \\ \Rightarrow \frac{{\tan }^3\theta -3\tan \theta }{1-3{\tan }^2\theta }=k\tan 3\theta \\ \Rightarrow -\tan 3\theta =k\tan 3\theta \Rightarrow k=-1\end{array}$