Find a Cartesian equation for the curve and identify it. $ r^2 \cos 2\theta = 1$

Question

Find a Cartesian equation for the curve and identify it. $$ r^2 \cos 2\theta = 1$$

I'm confused by the $2\theta.$

I isolated $r^2$ to get $r^2 = \frac{1}{\cos2\theta}$

Now, normally if it was just a $\cos \theta$ I would multiply both sides by $\frac{1}{r}$ and then substitute $r$ for $\sqrt {x^2+y^2}$ and then substitute $r\cos\theta$ for $x$. But the $2\theta$ doesn't allow for that to happen.

Answer 1

There are a few ways to approach, but here's a hint for how I'd do it: $\cos 2\theta = 2\cos^2\theta - 1$, and $x = r \cos\theta$, so $x^2 = \dots$, and so...

Answer 2

Since $\cos 2 \theta = 2 \cos^2 \theta -1$, then $$r^2 \cos 2 \theta = 2 r^2 \cos^2 \theta - r^2 = 2(r \cos \theta)^2 - r^2 = 2x^2 - (x^2 + y^2) = x^2 - y^2 ,$$ so your curve is $x^2 - y^2 = 1$ which is known to be a hyperbola having the lines $y = \pm x$ as asymptotes.