$\cot ^{-1}\left(2 \cos \left(2 \operatorname{cosec}^{-1}(\sqrt{2})\right)\right)=\ldots$

To evaluate $\cot^{-1}\left(2 \cos\left(2 \csc^{-1}(\sqrt{2})\right)\right)$, begin with the innermost term $\csc^{-1}(\sqrt{2})$, which has principal value $\frac{\pi}{4}$ since $\csc(\frac{\pi}{4}) = \sqrt{2}$.

Substituting gives $2 \csc^{-1}(\sqrt{2}) = 2 \times \frac{\pi}{4} = \frac{\pi}{2}$. Then $\cos\left(\frac{\pi}{2}\right) = 0$, so the argument becomes $2 \times 0 = 0$.

Finally, $\cot^{-1}(0) = \frac{\pi}{2}$, as $\cot(\frac{\pi}{2}) = 0$ within the principal range.

Final answer: $\frac{\pi}{2}$