If $\cot (x)=\frac{5}{11}$ (in Quadrant -1$)$, find $\cos (2 x)=$ 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 \(\cos (2 x)=\cos^2 (x)-\sin^2 (x)\), so once we have the values of \(\cos (x)\) and \(\sin (x)\), we can substitute them into this formula to find the value of \(\cos (2 x)\).

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 \(\cos (2 x)\), we find that \(\cos (2 x)\) 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 \(\cos (2 x)\) in fraction form is \(\boxed{-\frac{48}{73}}\).