If $\cot (x)=\frac{5}{11}$ (in Quadrant -1$)$, find $\cos (2x)=$
Give the answer in fraction form

Step 1 ：We are given that $\cot (x)=\frac{5}{11}$ and we know that $\cot (x)=\frac{\cos (x)}{\sin (x)}$. We can use this to find the values of $\cos (x)$ and $\sin (x)$.

Step 2 ：We also know that $\cos (2x)={\cos }^2(x)-{\sin }^2(x)$, so once we have the values of $\cos (x)$ and $\sin (x)$, we can substitute them into this formula to find the value of $\cos (2x)$.

Step 3 ：Using the given information, we calculate the values of $\sin (x)$ and $\cos (x)$ to be approximately -0.9103664774626047 and -0.4138029443011839 respectively.

Step 4 ：Substituting these values into the formula for $\cos (2x)$, we find that $\cos (2x)$ is approximately -0.6575342465753424.

Step 5 ：However, the question asks for the answer in fraction form. We can convert this decimal to a fraction.

Step 6 ：The value of $\cos (2x)$ in fraction form is $\boxed{{\displaystyle -\frac{48}{73}}}$.