The cosecant is given by the inverse of the sine function, and the cotangent function is given by the cosine over the sine.
So, first let's calculate the sine function, then the cosine, and finally the cotangent.
(since the angle is in Quadrant IV, the sine is negative, the cosine is positive and the cotangent is negative)
[tex]\begin{gathered} \csc (\theta)=\frac{1}{\sin (\theta)} \\ -\frac{\sqrt[]{638}}{22}=\frac{1}{\sin (\theta)} \\ \sin (\theta)=\frac{-22}{\sqrt[]{638}} \\ \\ \sin ^2(\theta)+\cos ^2(\theta)=1 \\ (-\frac{22}{\sqrt[]{638}})^2+\cos ^2(\theta)=1 \\ \frac{484}{638}+\cos ^2(\theta)=1 \\ \cos ^2(\theta)=\frac{638}{638}-\frac{484}{638} \\ \cos ^2(\theta)=\frac{154}{638} \\ \cos (\theta)=\frac{\sqrt[]{154}}{\sqrt[]{638}} \\ \\ \cot (\theta)=\frac{\cos (\theta)}{\sin (\theta)} \\ \cot (\theta)=\frac{\frac{\sqrt[]{154}}{\sqrt[]{638}}}{\frac{-22}{\sqrt[]{638}}} \\ \cot (\theta)=-\frac{\sqrt[]{154}}{22} \end{gathered}[/tex]
Therefore the value of cot(theta) is equal to -(√154)/22.
