Answer
$-\frac{1}{\sqrt 3}$
Work Step by Step
Period of $cot $ function is $\pi$
It means
$\cot \left( \alpha \right) =\cot \left( \alpha +\pi k\right) $ where $k$ is integer number
$\cot \left( -\dfrac {22\pi }{3}\right) =\cot \left( -\dfrac {22\pi }{3}+8\pi \right) =\cot \dfrac {2\pi }{3} $
$\Rightarrow \cot \left( -\dfrac {22\pi }{3}\right) = \cot \dfrac {2\pi }{3}=\dfrac {\cos \dfrac {2\pi }{3}}{\sin \dfrac {2\pi }{3}}=\dfrac {\cos \left( \pi -\dfrac {\pi }{3}\right) }{\sin \left( \pi -\dfrac {\pi }{3}\right) }=\dfrac {\cos \pi \cos \dfrac {\pi }{3}+\sin \pi \sin \dfrac {\pi }{3}}{\sin \pi \cos \dfrac {\pi }{3}-\cos \pi \sin \dfrac {\pi }{4}}=\dfrac {\left( -1\right) \times \cos \dfrac {\pi }{3}+0}{0-\left( -1\right) \times \sin \dfrac {\pi }{3}}=\dfrac {-\cos \dfrac {\pi }{3}}{\sin \dfrac {\pi }{3}}=\dfrac {-\dfrac {1}{2}}{\dfrac {\sqrt {3}}{2}}=-\dfrac {1}{\sqrt {3}} $
