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Cute functional equations

Source: 52nd Putnam 1991 Problem B2

November 17, 2020
functionfunctional equationalgebrasuperior algebra

Problem Statement

Define functions ff and gg as nonconstant, differentiable, real-valued functions on RR. If f(x+y)=f(x)f(y)−g(x)g(y)f(x+y)=f(x)f(y)-g(x)g(y), g(x+y)=f(x)g(y)+g(x)f(y)g(x+y)=f(x)g(y)+g(x)f(y), and f′(0)=0f'(0)=0, prove that (f(x))2+(g(x))2=1\left(f(x)\right)^2+\left(g(x)\right)^2=1 for all xx.
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