On Viète-like formulas

Abstract

The very first recorded use of an infinite product in mathematics is the so-called Viète’s formula, in which each of its factors contains nested square roots of 2 with plus signs inside. Concretely, it reads 2π=222+222+2+22⋯, and it can be proved by iterating the double angle formula sin2x=2cosxsinx, thus obtaining the infinite product 2/π=∏n=2∞cos(π/2n).This paper focuses, first, on the wide variety of iterations that the identity cosx=2cos((π+2x)/4)cos((π−2x)/4) admits; next, on the infinite products of cosines derived from these iterations and finally, on how these infinite products of cosines give rise to striking formulas.