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