Show that \(( \cos \theta + \sin \theta ) ^ { 2 } = 1 + \sin 2 \theta\).
A curve has cartesian equation
$$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 } = ( x + y ) ^ { 4 }$$
Given that \(r \geqslant 0\), show that the polar equation of the curve is
$$r = 1 + \sin 2 \theta$$
The curve with polar equation
$$r = 1 + \sin 2 \theta , \quad - \pi \leqslant \theta \leqslant \pi$$
is shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{f90167c3-2ffd-464a-b2d2-9f86a8d64887-3_389_611_1062_708}
Find the two values of \(\theta\) for which \(r = 0\).