A unique property of sine

The following trig identity looks like a mistake but is correct:

sin(x + y) sin(x − y) = (sin(x) + sin(y)) (sin(x) − sin(y))

It looks as if someone fallaciously expanded

sin(x + y) “=” sin(x) + sin(y)

and

sin(x − y) “=” sin(x) − sin(y).

Although both expansions are wrong, their product is correct. That is, for all x and y,

sin(x + y) sin(x − y) = sin²(x) – sin²(y).

Not only does the sine function satisfy this identity, it is almost the only function that does [1]. The only functions f that satisfy

f(x + y) f(x − y) = f²(x) − f²(y)

are f(x) = ax and f(x) = a sin(bx), given some mild regularity conditions on f. [2]

If f(x) satisfies the identity above, then so does

g(x) = a f(bx)

and so the identity could only characterize sine up to a change in amplitude and frequency.

You could think of the linear solution

f(x) = ax

as the limiting case of the solution

f(x) = ab sin(x/b)

as b goes to infinity. So you could include linear functions as sine waves with “infinite amplitude” and “zero frequency.”

In [1] the authors also prove that if we restrict f to be a real-valued function of a real variable, the only solutions are of the form ax, a sin(bx), and a sinh(bx).

The hyperbolic sine is only a new solution from the perspective of real numbers. Since

sinh(x) = −i sin(ix),

sinh was already included in a sin(bx) as the special case a = −i and b = i.

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[1] R. A. Rosenbaum and S. L. Segal. A Functional Equation Characterising the Sine. The Mathematical Gazette , May, 1960, Vol. 44, No. 348 (May, 1960), pp. 97-105.

[2] It is sufficient to assume f is an entire function, i.e. a complex function of a complex variable that is analytic everywhere. Rosenbaum and Segal give the weaker but more complicated condition that f is “on defined over all complex numbers, continuous at a point, bounded on every closed set, and such that the set of non-zero zeros of f is either empty or bounded away from zero.”