**Solution:**

**We know that sec**^{2}x-tan^{2}x = 1

**sec**^{2}x = 1+tan^{2}x

**sec**^{2}x = 1 + {x - (1/4x)}^{2}

**sec**^{2}x = 1 + x^{2} + 1/16x^{2} - 2*x*1/4x

**sec**^{2}x = 1 + x^{2} + 1/16x^{2} - 1/2

**sec**^{2}x = (x+1/4x)^{2}

**sec x = +- (x+1/4x)**

**So, sec x - tan x = x+1/4x - x+1/4x, or -x-1/4x - x +1/4x**

**sec x - tan x = 1/2x, or -2x**

**so, correct option is (A)**