Answer
$y=0$
Work Step by Step
Formulas relating rectangular $(x,y)$ and polar $(r,\theta)$ coordinates:
$ x=r\cos\theta$ and $ y=r\sin\theta$
$r^{2}=x^{2}+y^{2}$ and $\displaystyle \tan\theta=\frac{y}{x}$.
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$ \cos 2\theta=1\qquad$ ... double angle identity for cos,
$2\cos^{2}\theta-1=1$
$2\cos^{2}\theta=2$
$\cos^{2}\theta=1\qquad/\times r^{2}$
$r^{2}\cos^{2}\theta=r^{2}$
$(r\cos\theta)^{2}=r^{2}$
Substituting $ r\cos\theta$ and $r^{2}$, using the above formulas
$x^{2}=x^{2}+y^{2}\qquad/-x^{2}$
$0=y^{2}$
$y=0$
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Alternatively,
If $\cos 2\theta=1$, then using the unit circle,
$2\theta=0$
$\theta=0$
For $\theta=0$, tan is defined and equals 0, so we can write
$\displaystyle \tan\theta=\frac{y}{x}=0,$ that is,
$y=0$
