[tex]\sec x=-\dfrac92\implies\cos x=-\dfrac29[/tex]
[tex]\sin^2x+\cos^2x=1\implies\sin x=\pm\sqrt{1-\cos^2x}[/tex]
[tex]\sin x<0\implies\sin x=-\sqrt{1-\left(-\dfrac29\right)^2}=-\dfrac{\sqrt{77}}9[/tex]
[tex]\cot x=\dfrac{\cos x}{\sin x}[/tex]
[tex]\implies\cot x=\dfrac{-\frac29}{-\frac{\sqrt{77}}9}=\dfrac2{\sqrt{77}}[/tex]
