If $\tan 2\theta \tan \theta =1$, then the value of $\tan 2\theta$ is:

Given equation:
$$\tan 2\theta \tan \theta =1⟹\tan 2\theta =\frac{1}{\tan \theta }=\cot \theta$$

Using the trigonometric co-function identity $\cot \theta =\tan (90^{\circ }-\theta )$:

$$\tan 2\theta =\tan (90^{\circ }-\theta )$$

$$2\theta =90^{\circ }-\theta$$

$$3\theta =90^{\circ }⟹\theta =30^{\circ }$$

Now, calculate the value of $\tan 2\theta$:

$$\tan 2\theta =\tan (2\times 30^{\circ })=\tan 60^{\circ }=\sqrt{3}$$