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?