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

Step 1 ：We are given that \(\cot (x)=\frac{5}{11}\) and we are asked to find \(\cos (2 x)\).

Step 2 ：We know that \(\cot(x) = \frac{\cos(x)}{\sin(x)}\). From this, we can find the values of \(\cos(x)\) and \(\sin(x)\).

Step 3 ：Let's denote \(\cos(x)\) as 'a' and \(\sin(x)\) as 'b'. We can use the Pythagorean identity \(a^2 + b^2 = 1\) to find the values of 'a' and 'b'.

Step 4 ：By substituting \(\cot(x)\) into the equation, we get \(a = \frac{5}{\sqrt{5^2 + 11^2}}\) and \(b = \frac{11}{\sqrt{5^2 + 11^2}}\).

Step 5 ：Then, we can use the double angle formula for cosine, which is \(\cos(2x) = 1 - 2\sin^2(x)\) or \(\cos(2x) = 2\cos^2(x) - 1\), to find the value of \(\cos(2x)\).

Step 6 ：Substituting the values of 'a' and 'b' into the equation, we get \(\cos(2x) = 1 - 2*(\frac{11}{\sqrt{5^2 + 11^2}})^2\) or \(\cos(2x) = 2*(\frac{5}{\sqrt{5^2 + 11^2}})^2 - 1\).

Step 7 ：Solving the above equation, we get \(\cos(2x) = -0.5867768595041323\).

Step 8 ：So, the final answer is \(\boxed{-0.5867768595041323}\).