Given cot x=1/5, what is the value cos^2 x?

If cot x = 1/5, then tan x = 1/cot x =
5.

But tan x = sin x/cos x => sin x/cos x =
5

sin x = 5*cos x

We'll raise to square
both sides:

(sin x)^2 = 25*(cos
x)^2

We'll add (cos x)^2 both
sides:

(sin x)^2 + (cos x)^2 = 25*(cos x)^2 + (cos
x)^2

But, from Pythagorean identity, we'll
have:

(sin x)^2 + (cos x)^2 = 1

1 =
26*(cos x)^2

We'll divide by 26 and we'll use symmetric
property:

(cos x)^2 =
1/26

The value of (cos x)^2 is (cos x)^2 =
1/26.