Question

I need to convert the rectangular equation to polar form

Answer 1

You don't want 'x' in the polar form.  The arguments
of the sin and cos should be angles.

You're doing great.  You're already through the tough part.
You knew that  x = r cos(α)  and  y = r sin(α)  .
The correct equation for where you stopped is

                              x· y  =  r² · cos(α) · sin(α)  =  1

You can simplify it a little if you remember the 'double-angle' equation:

                             sin(2α)  =  2 · sin(α) · cos(α)

So      cos(α) · sin(α)  =  (1/2) sin(2α)

and              r² · cos(α) · sin(α)  =  1

                   r² · sin(2α)  =  2