Calculate: $\;$ $\cot^{-1} \left[\tan \left(- 37^{\circ}\right)\right]$
Given Expression (GE): $\;$ $\cot^{-1} \left[\tan \left(- 37^{\circ}\right)\right]$
$= \cot^{-1} \left[- \tan 37^\circ\right]$
$= \cot^{-1} \left[- \tan \left(90^\circ - 53^\circ\right)\right]$
$= \cot^{-1} \left[- \cot 53^\circ\right]$
Let $\;$ $\cot 53^\circ = x$
Then, GE $= \cot^{- 1} \left(- x\right) = \pi - \cot^{-1} x$
$= \pi - \cot^{-1} \left(\cot 53^\circ\right)$ $\;\;\;$ [substituting for $x$]
$= \pi - 53^\circ = 127^\circ$
Formulas used:
$\tan \left(- \theta\right) = - \tan \theta$
$\cot \theta = \tan \left(\dfrac{\pi}{2} - \theta\right)$
$\cot^{-1} \left(- \theta\right) = \pi - \cot^{-1} \theta, \;\;\; \forall \; \theta \in R$
$\cot^{-1} \left(\cot \theta\right) = \theta, \;\;\; \forall \; \theta \in \left(0, \pi\right)$