Proof the identity 1 + sin 2x/cos 2x = cos x + sin x/ cos x - sin x

The equation as you typed it is NOT an identity, you must have meant:

(1+sin2x)/(cos2x) = (cosx + sinx)/(cosx - sinx)

LS = (1 + 2sinxcosx)/(cos^2x - sin^2x)

RS = (cosx + sinx)/(cosx - sinx) * (cosx + sinx)/(cosx + sinx)
= (cos^2x + 2sinxcosx + sin^2x)/(cos^2x - sin^2x)
= (1 + 2sinxcosx)/(cos^2x - sin^2x
= LS