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

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

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 $\cos (2x)=1-2\ast (\frac{11}{\sqrt{5^2+{11}^2}}{)}^2$ or $\cos (2x)=2\ast (\frac{5}{\sqrt{5^2+{11}^2}}{)}^2-1$.

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

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