algebra precalculus – About the Nicomedes Conchoid

I’d be glad if you show me some light here.

It’s stated all over the net that the Nicomedes Conchoid polar equation is $r=asectheta + b $.

I’m trying to figure out that equation by locus points but in my construction I get $r=asectheta pm b$ (there’s two points in the intersection of the circle)

While graphing the equations apart with the plus sign and the minus sign, I get the same graph (the sign just seems to indicate the direction of the trace).

Here’s the question: is there a way to show that $r=asectheta + b $ and $r=asectheta – b $ are the same graph? Maybe doing some algebra to show that by cosine properties they trace the same path? So far, I got nothing trying that (is that even possible?)

I could try to show that both equations get the same cartesian equation, but is that enough to say that there’s no problem if we leave that minus sign behind?

Thanks in advance