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Show an equivalence between given sets in Cartesian and polar coordinates

by RoRi

Show that the sets of points satisfying

    \[ (x^2 + y^2)^2 = |x^2-y^2|; \qquad \qquad r = \sqrt{|\cos (2\theta)|}. \]

are equal.

Letting x = r \cos \theta and y = r \sin \theta we plug in to the given Cartesian equation,

    \begin{align*} (x^2 + y^2)^2 = | x^2 - y^2 | && \implies && (r^2 \cos^2 \theta + r^2 \sin^2 \theta)^2 &= | r^2 \cos^2 \theta - r^2 \sin^2 \theta| \\ && \implies && r^4 &= r^2 | \cos (2 \theta) | & (r^2 > 0, \text{ so } |r^2| = r^2)\\ && \implies && r^2 &= | \cos (2 \theta) | \\ && \implies && r &= \sqrt{ | \cos (2 \theta)|}. \end{align*}

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