Can someone help prove this identity

(tan x / 1 + sec x) + (1 + sec x / tan x) = 2csc x

Thanks!!!

Heres how it goes:
(tan x / 1 + sec x) + (1 + sec x / tan x) = 2csc x
I got a common denominator for the fractions so I can add them together:

(tan^2 x / tan x (1 + sec x)) + (1 + 2sec x + sec^2 x / tan x (1 + sec x)) = 2csc x
and then I added the two numerators:

(tan^2 x + 1 + 2sec x + sec^2 x) / (tan x (1 + sec x)) = 2csc x
I changed tan^2 x + 1 into sec^2 x (a trig identity):

(sec^2 x + 2sec x + sec^2 x) / (tan x (1 + sec x)) = 2csc x
You have 2 sec^2 x and a 2sec x, so I factored out a 2sec x:

(2sec x(1 + sec x)) / (tan x (1 + sec x)) = 2csc x
You can cancel out the (1+ sec x) on top and bottom:

2sec x / tan x = 2csc x
and since sec x / tan x = csc x, you have:

2csc x = 2csc x

If you write down these steps on paper, they may be easier to see!