Answer
The cartesian form of the expression is $x^{2}-y^{2}=1$ which is a unit hyperbola opening in the x-direction.
Work Step by Step
We are told that $r^{2}cos2\theta=1$.
Using the double angle identity for cosine, we get $r^{2}(cos^{2}\theta-sin^{2}\theta)=1$.
Distributing, we get $r^{2}cos^{2}\theta-r^{2}sin^{2}\theta=1$.
We can rewrite this as $(rcos\theta)^{2}-(rsin\theta)^{2}=1$.
Using the polar-to-cartesian substitutions of $x=rcos\theta$ and $y=rsin\theta$, we can rewrite the entire expression as $x^{2}-y^{2}=1$, which is a unit hyperbola opening in the x-direction.
