I'm looking for an algebraic way to calculate the $n$th value of infinite set $q^†$ where $q_nx^2$ intersects both $\sin(x)$ and $\cos(x)$ at the same point as $\sin(x)$ and $\cos(x)$ intersect. Is such a formula possible? I've already figured out the closest intersection to the origin — $q_0=\frac{8\sqrt{2}}{\pi^{2}}$, therefore intersecting both $\sin(x)$ and $\cos(x)$ at $\left(\frac{\pi}{4},\frac{\sqrt{2}}{2}\right)$ — but now, I'm curious as to if this is universally calculable.
$^†$This is parsing $0$ as the index for the intersection closest to the origin, allowing negative indices for solutions to the left and positive indices for solutions to the right. While this is admittedly nonstandard, I can't think of any other way to do it...