Now, let's find the expression for $\tan^{-1}(\cot x)$.
Since $\cot x = \frac{\cos x}{\sin x}$, we have $\tan^{-1}(\cot x) = \tan^{-1}\left(\frac{\cos x}{\sin x}\right)$.
Now, let's use the property of inverse trigonometric functions: $\tan(\tan^{-1}(x)) = x$.
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