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 $\theta$?

Continuing from @egreg ‘s work
$\sin ^2 \theta = \dfrac {x}{x + y}$ and $\cos ^2 \theta = \dfrac {y}{x + y}$
Then, $\dfrac {y}{x + y} - \dfrac {x}{x + y} = \cos ^2 \theta - \sin ^2 \theta $ 
$∴\dfrac {y - x}{y + x} = \cos 2\theta$
Construct the above angle and then bisect it.
Note that this method can effectively avoid the square root construction.
