Algebra seems like a blur, so I need a bit of help with this problem.(~)==theta

The original problem is this, and you are supposed to prove that it is true

cos(~)

------- = Sec(~)+tan(~)

1-sin(~)

my work:

1/sec * 1/1-sin(~)

1/sec(1-sin(~)

1/sec(~)-sec(~)*sin(~)

----This is where I got confused(I couldn't remember if you could use the inverse to make this statement true)------

(1/sec(~)-tan(~))^-1=Sec(~)+tan(~)

Could I get a yes or no if I did it right? Thanks