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Show an equivalence between given sets in Cartesian and polar coordinates

by RoRi

Show that the sets of points satisfying

    \[ (x^2 + y^2)^2 = x^2-y^2, \quad y^2 \leq x^2; \qquad \qquad r = \sqrt{\cos (2\theta)}, \quad \cos (2 \theta) \geq 0. \]

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) \\ && \implies && r^2 &= \cos (2 \theta) \\ && \implies && r &= \sqrt{\cos (2 \theta)}. \end{align*}

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