Calculus: Early Transcendentals 8th Edition

The cartesian form of the expression is $x^{2}-y^{2}=1$ which is a unit hyperbola opening in the x-direction.
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.