Question 1
(BUL 7) IMO 5. Let $f$ and $\varphi$ be real functions defined on the set $\mathbb R$ satisfying the functional equation
$$f(x+y)+f(x-y)=2\varphi(y)f(x),\tag{1}$$
for arbitrary real $x,y$ (give examples of such functions). Prove that if $f(x)$ is not identically $0$ and $|f(x)|\le1$ for all $x$, then $|\varphi(x)|\le1$ for all $x$.
$$f(x+y)+f(x-y)=2\varphi(y)f(x),\tag{1}$$
for arbitrary real $x,y$ (give examples of such functions). Prove that if $f(x)$ is not identically $0$ and $|f(x)|\le1$ for all $x$, then $|\varphi(x)|\le1$ for all $x$.