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**Reckoner** $\displaystyle r = \sqrt{1 + \cos2\theta}$

$\displaystyle \Rightarrow r = \sqrt{1 + \left(2\cos^2\theta - 1\right)}$

$\displaystyle \Rightarrow r = \sqrt{2\cos^2\theta} = \sqrt2\sqrt{\cos^2\theta} = \sqrt2\left\lvert\cos\theta\right\rvert$

$\displaystyle \Rightarrow r^2 = r\sqrt2|\cos\theta|\qquad\qquad\color{red}\text{Mu ltiplying both sides by }r$

$\displaystyle \Rightarrow r^2 = \sqrt2|r\cos\theta|\qquad\qquad\color{red}\text{Si nce }r\text{ is nonnegative, }r|\cos\theta| = |r||\cos\theta| = |r\cos\theta|$

$\displaystyle \Rightarrow x^2 + y^2 = \sqrt2|x|\qquad\qquad\color{red}\text{Substitution }$

$\displaystyle \Rightarrow y^2 = \sqrt2|x| - x^2$