An electrostatic “proof” that cos 36°−cos 72°=1/2

Using methods of electrostatics, we derive an interesting identity connected to geometry, the golden ratio, and exact trigonometric values: cos 36°−cos 72°=1/2. We analyze the electric field at the center of a pentagon due to charges on four of its vertices using two different methods: (1) vector addition of the fields from each charge, and (2) superposition with a rotationally symmetric variant of the setup. By equating the expressions produced by both methods, we obtain the above identity. Generalizing this procedure to a (2N+1)-sided polygon, we further show that the trigonometric series ∑n=1N cos [π(2n−1)/(2N+1)]=1/2.