the sine product formula
The sine function, viewed in the complex plane, is entire and vanishes exactly at the integers. The fundamental theorem of algebra suggests that a function should be the product of factors that vanish at its roots — and Euler dared to write sine as an infinite such product, two centuries before Weierstrass justified it. The result is the sine product formula: sin(pi z) = pi z times the product over n >= 1 of (1 - z^2/n^2). Every factor 1 - z^2/n^2 vanishes at z = +-n, the leading pi z supplies the zero at the origin, and together they account for every zero of sine.
Read against the construction machinery, this is the canonical Weierstrass product for the zero set {all integers}. The zeros are at n != 0; the smallest degree making sum 1/|n|^(p+1) converge is p = 1, so the canonical elementary factors are E_1(z/n), which combine in conjugate pairs (z = n and z = -n) into the real-looking factor 1 - z^2/n^2. No e^g correction beyond the constant pi is needed, because the symmetry of the zeros keeps the product convergent on its own. So the formula is not a lucky coincidence; it is exactly what Weierstrass's theory produces for the integers.
The formula is a generator of classical identities. Taking the logarithmic derivative (differentiate log of both sides) turns the product into the cotangent partial-fraction expansion pi cot(pi z) = 1/z + sum 2z/(z^2 - n^2). Setting z = 1/2 collapses the product into the Wallis product pi/2 = (2/1)(2/3)(4/3)(4/5) ... for pi. And it underlies the reflection formula for the gamma function, Gamma(z) Gamma(1 - z) = pi / sin(pi z), tying this construction to the special functions. Euler's bold infinite product turns out to be a keystone.
Set z = 1/2 in sin(pi z) = pi z product (1 - z^2/n^2): the left side is sin(pi/2) = 1, so 1 = (pi/2) product over n >= 1 of (1 - 1/(4n^2)). Rearranging gives Wallis's product pi/2 = product over n >= 1 of (4n^2)/(4n^2 - 1) = (2*2)/(1*3) * (4*4)/(3*5) * ... — a formula for pi extracted directly from the sine product.
Evaluating the sine product at z = 1/2 yields Wallis's product for pi/2.
The factors must be grouped as 1 - z^2/n^2 (pairing +n with -n), not summed as separate (1 - z/n) terms: the unpaired product diverges, while the paired one converges absolutely. The convergence lives in the symmetric grouping.