You need to transform both sides expanding the function sec x into fraction such that:
`1/(1/(cos x) - tan x) = 1/(cos x)+ tan x`
Cross multiplying yields:
`1 = (1/(cos x) - tan x)(1/(cos x)+ tan x)`
You may transform the special product to the right in difference of squares such that:
`1 = 1/(cos^2 x) - tan^2 x`
This last line expresses one of the three forms of basic trigonometric identity, hence the identity `1/(sec x-tan x)=sec x+tan x` is established.