What was this theorem called

Solution 1:

Write $E(x) = C(x)+i S(x)$. Then $E(x+y)=E(x)E(y)$. This is a multiplicative variant of Cauchy's functional equation. Without further hypotheses on $S$ and $C$ it is likely that there are many solutions.

If $S$ and $C$ are assumed differentiable, then $E$ satisfies $E'=E$. If moreover $E$ is not identically zero, then $E(0)=1$ and so $E=\exp$. By Euler's formula, $S=\sin$ and $C=\cos$.

So, one answer to your question is uniqueness of the exponential function from its differential equation.

Solution 2:

Hint $\:$ Said addition laws are true iff $\rm\: E(X) = C(X) + {\it i}\: S(X)\:$ satisfies $\rm\:E(X+Y)\: =\: E(X)\:E(Y)\:.\:$

Solution 3:

I found an article that seems directly relevant to the solution of the equations at: Dr. W Harold Wilson 'On Certain Related Functional Equations' AMS (read Dec 27,1917) Hope this is useful.