Given a formula for a polar equation: $$\ r^2 = a^2 \cos^22 \theta $$
It could be said that to find the points parallel to the initial line, $$\frac{dy}{dx} = \frac{d (r\sin\theta)}{d\theta} = 0$$
As $$y^2 = r^2 \sin^2\theta$$
It can be said that $$y^2 = a^2 (1-2\sin^2\theta)(\sin^2\theta) $$ $$y^2 = a^2 (\sin^2\theta - 2\sin^4\theta) $$
So to now find the points at which it is parallel to the initial line, I assumed that the square root of $y^2$ should be taken before it is differentiated and equated to $0$.
ie: $$ y = a \sqrt{ (\sin^2\theta - 2\sin^4\theta)} $$ then $$ \frac{d (a \sqrt{ (\sin^2\theta - 2\sin^4\theta)})}{d\theta} = 0 $$
and solve as such.
However after looking at the answers, the question was meant to be solved by differentiating $y^2$ and setting that to $0$ instead:
I cant figure out why the solution uses $$\frac{d (y^2)}{d\theta} = 0$$ rather than: $$\frac{d y}{d\theta} = 0$$
It's done this way for ease of calculation. $y$ will be maximum when $y^2$ is maximum.