ILL 1978 Problem 28

Let $c, s$ be real functions defined on $\mathbb{R}\setminus\{0\}$ that are nonconstant on any interval and satisfy $$c\left(\frac{x}{y}\right)= c(x)c(y) - s(x)s(y)\text{ for any }x \neq 0, y \neq 0$$ Prove that: $(a) c\left(\frac{1}{x}\right) = c(x), s\left(\frac{1}{x}\right) = -s(x)$ 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 $c(x) + s(x) = x^n$ for all $x$ , where $n$ is a given positive integer.