If $\cot (x)=\frac{5}{11}$ (in Quadrant-I), find $\cos (2 x)=$

Step 1 ：We are given that \(\cot(x) = \frac{5}{11}\) in Quadrant-I.

Step 2 ：We can express \(\cos(2x)\) in terms of \(\sin(x)\) and \(\cos(x)\) as \(\cos(2x) = 1 - 2\sin^2(x)\) or \(\cos(2x) = 2\cos^2(x) - 1\).

Step 3 ：From the given \(\cot(x) = \frac{5}{11}\), we can find \(\cos(x)\) and \(\sin(x)\) using Pythagorean identity \(\sin^2(x) + \cos^2(x) = 1\).

Step 4 ：By substituting the values of \(\sin(x)\) and \(\cos(x)\) into the formula for \(\cos(2x)\), we can find the answer.

Step 5 ：Calculating the values, we get \(\cos(x) = 0.413802944301184\) and \(\sin(x) = 0.9103664774626048\).

Step 6 ：Substituting these values into the formula \(\cos(2x) = 2\cos^2(x) - 1\), we get \(\cos(2x) = -0.6575342465753427\).

Step 7 ：So, the final answer is \(\cos(2x) = \boxed{-0.6575342465753427}\).