You need to remember that `sec 2x = 1/ cos 2x` , hence, you should use the following trigonometric identity such that:

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

You need to find `tan 2x`  using the following formula such that:

`tan 2x = (2 tan x)/(1 - tan^2 x)`

The problem provides the value of tan x, hence, substituting 2 for tan x yields:

`tan 2x = (2*2)/(1 - 2^2) => tan 2x = -4/3`

You need to substitute `-4/3`  for `tan 2x ` in equation above such that:

`1 + (-4/3)^2 = 1/(cos^2 2x) => 1 + 16/9 =1/(cos^2 2x)`

`25/9 =1/(cos^2 2x) => 1/(cos 2x) = +-5/3`

Hence, evaluating the secant of double angle, using the provided information, yields `sec 2x = +-5/3.`