To find the value of cos x, we can use the given equation sec x tan x = 2.

Let's start by simplifying the equation.

Recall that sec x is the reciprocal of cos x, so we can rewrite the equation as:

(1/cos x) * (sin x/cos x) = 2

Now, let's multiply both sides of the equation by cos x to eliminate the fractions:

sin x = 2cos^2 x

Next, recall the trigonometric identity sin^2 x + cos^2 x = 1. We can rewrite the equation as:

sin x = 2(1 - sin^2 x)

Expanding the equation, we have:

sin x = 2 - 2sin^2 x

Rearranging the equation, we get:

2sin^2 x + sin x - 2 = 0

This is a quadratic equation in terms of sin x. Let's solve it using factoring.

We can rewrite the equation as:

(2sin x - 1)(sin x + 2) = 0

Setting each factor equal to zero, we have two possible solutions:

2sin x - 1 = 0 or sin x + 2 = 0

Solving the first equation, we get:

2sin x = 1

sin x = 1/2

To find the value of cos x, we can use the Pythagorean identity sin^2 x + cos^2 x = 1.

Substituting sin x = 1/2, we have:

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

1/4 + cos^2 x = 1

cos^2 x = 3/4

Taking the square root of both sides, we get:

cos x = ±√(3/4)

Since cosine is positive in the first and fourth quadrants, we take the positive square root:

cos x = √(3/4)

Simplifying, we have:

cos x = √3/2

Therefore, the correct answer is option D) 4/5.