Trigonometric proving help?

(sin 2x +1) / cos 2x
=(2 sin x cos x +1) / ( cos^2 x - sin^2 x)
divide the numerator and denominator by cos^2 x
=(2 tan x + sec^2 x) / (1 -tan^2 x)
= (2 tan x + 1+tan^2 x) / (1-tan^2 x)
= (tan^2 x + 2 tan x +1) /((1-tan x)(1+tan x))
= (tan x +1)^2 /((1+tan x)(1-tan x))
= (1+tan x) /(1-tan x)

Identities used :
sin2x = 2 cosx sinx
cos²x + sin²x = 1
cos2x = cos²x - sin²x = (cosx + sinx)(cosx - sinx)
cos²x + 2 cosx sinx + sin²x = (cosx + sinx)²
tanx = sinx/cosx

L.H.S.
= (sin2x + 1) / cos2x
= [2 cosx sinx + (cos²x + sin²x)] / (cos²x - sin²x)
= (cos²x + 2 cosx sinx + sin²x) / (cos²x - sin²x)
= (cosx + sinx)² / [(cosx + sinx)(cosx - sinx)]
= (cosx + sinx) / (cosx - sinx)
= [(cosx + sinx) / cosx] / [(cosx - sinx) / cosx]
= [(cosx/cosx) + (sinx/cosx)] / [(cosx/cosx) - (sinx/cosx)]
= (1 + tanx) / (1 - tanx)
= R.H.S.

Hence, (sin2x + 1) / cos2x = (1 + tanx) / (1 - tanx)