If sec θ = x + 1 / 4x, then prove that sec θ + tan θ = 2x or 1 / 2x

We need to consider two different cases.

Answer: The value of sec θ + tan θ = 2x or 1/2x as proved below.

We can make use of the trigonometric identity.

Explanation: 

Given, sec θ = x + 1 / 4x

Using the trigonometric identity, 1 + tan² θ = sec² θ

==> tan² θ = sec² θ - 1

Using sec² θ = (x + 1/4 x)², we get

==> 1 + tan² θ = x² + 1/16 x² + 1/2

tan² θ =  x² + 1/16 x² - 1/2 = (x - 1/4x)²

Taking square root on both sides, we get tan θ = ± x - 1/4x

(i) if tan θ = (x - 1/4x) and sec θ = (x + 1/4x), we have tan θ + sec θ = 2x ,

(ii) if tan θ = - (x - 1/4x) and sec θ = (x + 1/4x), we have tan θ + sec θ = 1/4x + 1/4x = 1/2x

Hence proved that the value of sec θ + tan θ is 2x or 1/2x.