 Solution

Apply the multivariable chain rule, keeping the other new coordinate fixed in each differentiation. At fixed $\theta$, the derivatives of $x,y$ with respect to $r$ are $\cos 2\theta ,\sin 2\theta$. At fixed $r$, their derivatives with respect to $\theta$ are $-2r\sin 2\theta ,2r\cos 2\theta$. ThusThe factor two arises because the ordinary polar angle is $2\theta$.