## Calculus (3rd Edition)

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