Identifying the symmetry of a polar function

Question

What is the purpose of $1+\cos(-\theta) \neq -r$

Answer 1

It looks like they check symmetry about the $y$-axis in two different ways:

• by verifying if $r(-\theta)=-r(\theta)$;
• by verifying if $r(\pi-\theta)=r(\theta)$.

What is the purpose of $1+\cos(-\theta) \neq -r$

So this refers to the first of the two above.

Side note: where is the argument about the symmetry with respect to the origin? They conclude "therefore not symmetric about the origin" but I would expect something like verifying whether $r(\theta+\pi) = r(\theta)$ to arrive at that conclusion.