I have the following equation which I am trying to solve,
$$ \frac{x^2}{y^2}- \frac{x^2}{y^4} -\frac{1}{2} \leq 0$$
Can anyone think of a way of simplifying the above, I don't think this is a form of quadratic equation but I want to simplify it and have a proper relationship between $x$ and $y$
Not sure if this is what you want, but you can write for $ y \ne 0$:$$\frac{x^2}{y^2}-\frac{x^2}{y^4}\le \frac12 \iff 2x^2(y^2-1 ) \le y^4 \implies \begin{cases} 2x^2 \le \dfrac{y^4}{y^2-1} && y^2 > 1 \\x \in \mathbb R&& y^2 \le 1\end{cases}$$
The first case can also be written as $|x| \le \dfrac{y^2}{\sqrt{2(y^2-1)}}$, for $|y| > 1$.