tan (x/2 + Π/4) = sec x + tan x

Here is my attempt:

Substituting 1 for tan (Π/4) and applying the tangent of a sum identity gives

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

Using the half angle identity for tangent, tan(x/2) = sin x /(1 + cos x) and simplifying the complex fraction gives

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

I do not know where to go from here. Please help!