sec(x)-cos(x)/tan(x)=sin(x)

The equation is incorrect as written. I think you actually meant this:

(sec(x)-cos(x))/tan(x) = sin(x)

To prove this simply replace sec(x) with 1/cos(x) and tan(x) with sin(x)/cos(x). Then you can apply the idfentity 1-cos^2(x) = sin^2(x), and it falls right out.