What's wrong with the calculation with polar coordinates here? Suppose $x=r\cos t$ and $y=r\sin t$. I did the following calculation:
$$
\begin{align}
&x^4+y^4=(r\cos t)^4+(r\sin t)^4\\
=&r^4(\sin^4t+\cos^4t)\\
=&r^4[(\sin^2t+\cos^2t)^2-2\sin^2t\cos^2t]\\
=&r^4(1-2\sin^2t\cos^2t)=r^4-2r^4\sin^2t\cos^2t
\end{align}
$$
On the other hand
$$
\begin{align}
x^4+y^4&=(x^2+y^2)^2-2x^2y^2\\
&=r^4-2r^2\sin^2t\cos^2t
\end{align}
$$
What is wrong with the calculation?

Notice that $2x^2y^2 = 2(r\sin t)^2(r\cos t)^2 = 2r^4\sin^2t\cos^2t$.
