Math Bite: A Generalization of a Curiosity that Feynman Remembered All His Life

W.A. BEYER(beyer@Lanl.GOV), J.D. LOUCK(jimlouck@Lanl.GOV) and D. ZEILBERGER(zeilberg@math.temple.edu)

If a boy named Morrie Jacobs told him that the cosine of 20 degrees multiplied by the cosine of 40 degrees multiplied by the cosine of 80 degrees equaled exactly one-eighth, he would remember that curiosity for the rest of his life, and he would remember that he was standing in Morrie’s father’s leather shop when he learned it. — James Gleick

[‘Genius’, p. 47]

Morrie Jacobs’s identity is the special case k = 3, a = 20^{o}, of the following identity that follows by induction on k using sin 2b = 2 sin b cos b, with b = 2^{k−1}a.

2^{k} ∏_{j=0}^{k−1} cos(2^{j}a) = \frac{sin(2^{k}a)}{sin a} .