if the

$\tan ^{-1}(\cot \theta)=2 \theta$

$\Rightarrow \tan ^{-1}\left[\tan \left(\frac{\pi}{2}-\theta\right)\right]=2 \theta$

$\Rightarrow \frac{\pi}{2}-\theta=2 \theta$

$\Rightarrow 3 \theta=\frac{\pi}{2}$

$\Rightarrow \theta=\frac{\pi}{6}$

Thus, the value of $\theta$ is $\frac{\pi}{6}$.

If $\tan ^{-1}(\cot \theta)=2 \theta$, then $\theta=\frac{\pi}{6}$