How do I solve 1/(sec(x))-(tan(x))

I don't understand how to get rid of the subtraction sign.

1/[sec(x) - tan(x)] * [sec(x) + tan(x)]/[sec(x) + tan(x)]

[sec(x) + tan(x)] / [sec²(x) - tan²(x)]

sin² + cos² = 1
tan² + 1 = sec²
sec² - tan² = 1

sec(x) + tan(x)

........1...............
---------------------
sec(x) - tan(x)...

........1
=------------------
...1..........sin(x)
---------- - -----------
.cos(x).....cos(x)

.........1
=-------------------
....1-sin(x)
...---------------
......cos(x)

............cos(x)
= 1 x ----------------
..........1-sin(x)

....cos(x).............. cos(x)
=---------------- or - -------------- answer//
...1- sin(x)...........sin(x) - 1

.

1/(sec(x) - tan(x))

1/(1/cos(x) - sin(x)/cos(x))

= cos(x)/(1 - sin(x))

= cos(x)(1 + sin(x))/cos²(x)

= sec(x) + tan(x)

1/(sec(x))-(tan(x))
= [ sec^2 x -- tan^2 x] / [sec x -- tan x]
= sec x + tan x