Can you show me, step by step, how to differentiate (1-secx)/(1+secx)

I got very messy looking answers when I tried it. My graphing calculators answer was very different from my own.

I'm going to start by converting everything to cos, since the derivative will be easier to work with.

Numerator: 1 - sec x = 1 - 1/cos x = (cos x - 1) / cos x
Denominator: 1 + sec x = 1 + 1 / cos x = (cos x + 1) / cos x

[(cos x - 1) / cos x] / [(cos x + 1) / cos x]
[(cos x - 1) / cos x] * [cos x / cos x + 1]
(cos x - 1) / (cos x + 1)

Quotient rule:
f = cos x - 1
f ' = -sin x
g = cos x + 1
g' = -sin x

d/dx (f / g) = (g*f ' - f * g') / g^2
((cos x + 1)(- sin x) - (cos x - 1)(-sin x) ) / (cos x + 1)^2
- cos x sin x - sin x + cos x sin x - sin x / (cos x + 1)^2
= -2 sin x / (cos x + 1)^2

Edited to add:

A textbook or computer program may give you - 2 sec x tan x / (1 + sec x)^2. This is equivalent to the answer I got:

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