You also may use the following trigonometric identities, such that:

`cot 2x = (cos 2x)/(sin 2x)`

`1 + cot^2 x = 1/sin^2 x`

The problem provides the value of `cot x` , such that:

`1 + (1/3)^2 = 1/sin^2 x => 10/9 = 1/sin^2 x => sin^2 x = 9/10`

`cos x = sqrt(1 - sin^2 x) => cos x = sqrt(1 - 9/10) => cos x = sqrt10/10`

`sin 2x = 2 sin x*cos x => sin 2x = 2*3sqrt10/10*sqrt10/10 => sin 2x = 6/10 => sin 2x = 3/5`

`cos 2x = 1 - 2sin^2 x => cos 2x = 1 - 18/10 => cos 2x = -8/10`

`cos 2x = -4/5`

Replacing `3/5` for `sin 2x` and `-4/5` for `cos 2x` yields:

`cot 2x = (-4/5)/(3/5) => cot 2x = -4/3`

**Hence, evaluating the value of cotangent cot x, using trigonometric identities, yields **`cot 2x = -4/3.`

We'll write cot 2x as a sum of 2 like angles.

cot 2x = cot (x+x)

cot 2x = [(cot x)^2 - 1]/(cot x+ cot x)

cot 2x = [(cot x)^2 - 1]/(2cot x)

We know, from enunciation, that cot x= 1/3.

We'll substitute the value of cot x in the formula for cot 2x.

cot 2x = [(1/3)^2 - 1]/(2*1/3)

cot 2x =(-8/9)*(3/2)

cot 2x = -8/6

cot 2x = -4/3