Using the fact that [tex]x = r \cos \theta[/tex] and [tex]y = r \sin \theta[/tex], we can write the Cartesian equation as
[tex]x^4 - y^4 = 16[/tex]
[tex](r \cos \theta)^4 - (r \sin \theta)^4 = 16[/tex]
[tex]r^4 (\cos^4 \theta - \sin^4 \theta) = 16[/tex]
[tex]r^4 (\cos^2 \theta + \sin^2 \theta) (\cos^2 \theta - \sin^2 \theta) = 16[/tex]
Since [tex]\sin^2 \theta + \cos^2 \theta = 1[/tex] and [tex]\cos 2\theta = \cos^2 \theta - \sin^2 \theta[/tex],
[tex]r^4 (1) (\cos 2\theta) = 16[/tex]
[tex]\bf r^4 = \frac{16}{\cos 2\theta}[/tex]
