Chapter 12 - Parametric Equations, Polar Coordinates, and Conic Sections - 12.3 Polar Coordinates - Exercises - Page 619: 39

$$r^2=\sec (2\theta).$$

We have $$x^2-y^2=1\Longrightarrow r^2\cos^2\theta -r^2\sin^2\theta=1\\ \Longrightarrow r^2(\cos^2\theta -\sin^2\theta)=1\\ \Longrightarrow r^2\cos(2\theta )=1. $$ Hence, the polar equation of the hyperbola is $$r^2=\sec (2\theta).$$