Proof:

Let's start by using the given information: sec(A) = x and tan(A) = 1/4x.

We want to prove that sec(A) * tan(A) = 2x * 1/2x.

Using trigonometric identities:
We know that sec(A) = 1/cos(A) and tan(A) = sin(A)/cos(A).

So, we can rewrite sec(A) * tan(A) as (1/cos(A)) * (sin(A)/cos(A)).

Simplifying the expression:
To simplify this expression, we can multiply the numerators and denominators together.

(1 * sin(A)) / (cos(A) * cos(A))

Using the trigonometric identity:
We can apply the Pythagorean identity, which states that sin^2(A) + cos^2(A) = 1.

From this identity, we can derive that sin(A) = √(1 - cos^2(A)).

So, we can substitute sin(A) with √(1 - cos^2(A)) in the expression.

(1 * √(1 - cos^2(A))) / (cos(A) * cos(A))

Simplifying further:
Let's simplify the expression by multiplying the numerator and denominator.

√(1 - cos^2(A)) / (cos^2(A))

Using the trigonometric identity:
We can use the identity sec^2(A) = 1 + tan^2(A), which implies that cos^2(A) = 1 / (1 + tan^2(A)).

Substituting this value in the expression, we get:

√(1 - (1 / (1 + tan^2(A)))) / (1 / (1 + tan^2(A)))

Simplifying the expression:
To simplify further, we can multiply the numerator and denominator by (1 + tan^2(A)).

√((1 + tan^2(A)) - 1) / 1

√(tan^2(A)) / 1

Simplifying the expression:
The square root of tan^2(A) is simply tan(A).

So, the expression simplifies to:

tan(A)

Conclusion:
After simplifying the expression, we have shown that sec(A) * tan(A) equals tan(A).

Therefore, we have proved that sec(A) * tan(A) = 2x * 1/2x.