Eliminate $t$ from the equations: $x = \frac{1}{t} - t \, , \, y = \frac{1}{t^2} - 1$ $$x = \frac{1}{t} - t \tag{1}$$
$$y = \frac{1}{t^2} - 1 \tag{2}$$
$$\frac{1}{t^2} = y + 1$$
$$t^2 = \frac{1}{y + 1}$$
$$t = \pm\sqrt{\frac{1}{y + 1}}$$
$$x = \frac{1 - t^2}{t}$$
$$y = \frac{1 - t^2}{t^2}$$
$$\frac{x}{y} = \frac{1 - t^2}{t} \times \frac{t}{1 - t^2}$$
$$\frac{x}{y} = t \tag{3}$$
Substitute $(3)$ into $(1)$
$$x = \frac{y}{x} - \frac{x}{y}$$
$$x = \frac{y^2 - x^2}{xy}$$
$$x^2y = y^2 - x^2$$
$$x^2y + x^2 = y^2$$
$$x^2(y + 1) = y^2$$
$$x^2 = \frac{y^2}{y + 1}\tag{4}$$
$$x = \sqrt{\frac{y^2}{y + 1}}$$
From $(4):$
$$x^2y + x^2 = y^2$$
$$y^2 - x^2y = x^2$$
$$y(y-x^2) = x^2$$
I have no idea how to progress from that point. I've tried different methods in paper to make $y$ the subject of the formula, but none of them were fruitful.
How do I make $y$ the subject of the formula in $(4)$?

Assuming that the manipulations up till (4) are correct,
$$y^2 - x^2y=x^2$$
$$(y-\frac{x^2}{2})^2-\frac{x^4}{4}=x^2$$
$$(y-\frac{x^2}{2})^2=x^2+\frac{x^4}{4}$$
$y=\frac{x^2}{2}+\sqrt{x^2+\frac{x^4}{4}}$ or $y=\frac{x^2}{2}-\sqrt{x^2+\frac{x^4}{4}}$
