Answer
The value of the expression is $-\frac{5\sqrt{11}}{11}$
Work Step by Step
We have the provided expression as
$\cot \left[ {{\cos }^{-1}}\left( -\frac{5}{6} \right) \right]$.
Let us assume
$\begin{align}
& \left[ {{\cos }^{-1}}\left( -\frac{5}{6} \right) \right]=\theta \\
& \cos \theta =\frac{-5}{6}
\end{align}$
Then, $\theta $ lies in the 2nd quadrant.
And the expression reduces to $\cot \theta $
$\cot \theta =\frac{\cos \theta }{\begin{align}
& \sqrt{1-{{\cos }^{2}}\theta } \\
& =\frac{\frac{-5}{6}}{\sqrt{1-\frac{25}{36}}} \\
& =\frac{-5\sqrt{11}}{11} \\
\end{align}}$
Hence, $\cot \left[ {{\cos }^{-1}}\left( -\frac{5}{6} \right) \right]=\frac{-5\sqrt{11}}{11}$.
