IMO 1978 LL GDR28

Origin: GDR

Problem

Let c, s be real functions defined on R{0} that are nonconstant on any interval and satisfy c x y  = c(x)c(y) −s(x)s(y) for any x ̸= 0, y ̸= 0. Prove that: (a) c(1/x) = c(x), s(1/x) = −s(x) for any x ̸= 0, and also c(1) = 1, s(1) = s(−1) = 0; (b) c and s are either both even or both odd functions (a function f is even if f(x) = f(−x) for all x, and odd if f(x) = −f(−x) for all x). Find functions c, s that also satisfy c(x) + s(x) = xn for all x, where n is a given positive integer.