How to verify (1/1-sin x)-(1/1+sin x)= 2sec x tan x

Should be shown as :-

1 / (1 - sin x) - 1 / (1 + sin x)

1 + sin x - (1 - sin x)
---------------------------
(1 - sin x)(1 + sin x )

2 sin x
--------------------------
(1 - sin x)(1 + sin x)

2sinx
-----------------
1 - sin²x

2 sinx
------------------
cos x cos x

2 tan x sec x

(1/1-sin x)-(1/1+sin x) = [(1+sin x) - (1 - sin x)]/[(1 - sin x)(1 + sin x)]

= 2(sin x)/[1 - sin^2 x] = 2(sin x)/cos^2 x = 2[1/cos x]*[sin x/cos x] = 2 sec x tan x.

1 / (1 - sin(x)) - 1 / (1 + sin(x)) =>
(1 + sin(x) - 1 * (1 - sin(x))) / ((1 - sin(x)) * (1 + sin(x))) =>
(1 + sin(x) - 1 + sin(x)) / (1 - sin(x)^2) =>
2 * sin(x) / cos(x)^2 =>
2 * (sin(x)/cos(x)) * (1/cos(x)) =>
2 * tan(x) * sec(x)