Answer
$$ x^{2}-y^{2}=1$$
Hyperbola centered at $(0,0)$ with foci on the $x-$axis
Work Step by Step
Since
\begin{align*}
r^{2} \cos 2 \theta&=1 \\
r^{2}\left(\cos ^{2} \theta-\sin ^{2} \theta\right)&=1 \\
(r \cos \theta)^{2}-(r \sin \theta)^{2}&=1 \\
x^{2}-y^{2}&=1
\end{align*}
Hyperbola centered at $(0,0)$ with foci on the $x-$axis