Answer
$$r = 2\sqrt {\sec 2\theta } $$
Work Step by Step
$$\eqalign{
& {x^2} - {y^2} = 4 \cr
& {\text{Convert to polar form, using }}x = r\cos \theta ,{\text{ }}y = r\sin \theta \cr
& {\left( {r\cos \theta } \right)^2} - {\left( {r\sin \theta } \right)^2} = 4 \cr
& {r^2}{\cos ^2}\theta - {r^2}{\sin ^2}\theta = 4 \cr
& {r^2}\left( {{{\cos }^2}\theta - {{\sin }^2}\theta } \right) = 4 \cr
& {\text{Use the identity }}{\cos ^2}\theta - {\sin ^2}\theta = \cos 2\theta \cr
& {r^2}\cos 2\theta = 4 \cr
& {r^2} = 4\sec 2\theta \cr
& r = 2\sqrt {\sec 2\theta } \cr
& {\text{The equation represents a hyperbola}} \cr
& {\text{Graph}} \cr} $$
