Answer
$$(x^2+y^2)^2=x^2-y^2.$$
Work Step by Step
Since $r^2=\cos (2\theta)$, then we have
$$r^2=\cos^2\theta-\sin^2\theta=\frac{x^2}{r^2}-\frac{y^2}{r^2}.$$
Hence, $r^4=x^2-y^2$, which can be written in the form
$$(x^2+y^2)^2=x^2-y^2.$$
Calculus (3rd Edition)
by Rogawski, Jon; Adams, Colin
Published by
W. H. Freeman
ISBN 10:
1464125260
ISBN 13:
978-1-46412-526-3
Chapter 12 - Parametric Equations, Polar Coordinates, and Conic Sections - 12.3 Polar Coordinates - Exercises - Page 619: 40
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