$x^{2/3} + y^{2/3} = a^{1/3} \Rightarrow 2/3 x^{-1/3} x_1 + 2/3 \cdot y^{-1/3} = 0 \Rightarrow x^{-1/3} x_1 + y^{-1/3} = 0 \Rightarrow -x_1 = \frac{y^{-1/3}}{x^{-1/3}}$
$\frac{x^{1/3}}{\sin\theta} = \frac{y^{1/3}}{\cos\theta} = \sqrt{x^{2/3}+y^{2/3}} = \sqrt{a^{2/3}} = a^{1/3}$
$x = a\sin^3\theta;\; y = a\cos^3\theta$
$y - a\cos^3\theta = \tan\theta(x - a\sin^3\theta)$ $\Rightarrow y\cos\theta - a\cos^4\theta = \sin\theta(x - a\sin^3\theta)$ $\Rightarrow y\cos\theta - x\sin\theta = a(\cos^2\theta + \sin^2\theta)(\cos^2\theta - \sin^2\theta) = a\cos 2\theta$