If $tan^2 \theta = \frac{x}{y}$ how can we construct the angle $\theta$? If we are given the values of $x$ and $y$ and we know that $\tan^2 \theta = \dfrac{x}{y}$ is it possible for us to construct the angle $\theta$?

Set $z=x/y$, for simplicity. Then
$$
z=\frac{\sin^2\theta}{\cos^2\theta}=\frac{1}{\cos^2\theta}-1
$$
so
$$
\cos^2\theta=\frac{1}{z+1}=\frac{y}{x+y}
$$
and
$$
\sin^2\theta=1-\frac{1}{z+1}=\frac{z}{z+1}=\frac{x}{x+y}
$$
Depending on which quadrant $\theta$ lives in, you can choose the signs in
$$
\cos\theta=\pm\sqrt{\frac{y\vphantom{A}}{x+y}},
\quad
\sin\theta=\pm\sqrt{\frac{x\vphantom{A}}{x+y}}
$$
