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*) = *f*²(*x*) − *f*²(*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*.

***

[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.”

This woke my brain this morning better than the first mug of coffee. Thanks!

I was aware of the product identity, but oblivious to its uniqueness.