Question 190
Let $f, g: \mathbb{R} \to \mathbb{R}$ be continuous, non-constant functions satisfying
$$
f(x - y) = f(x)f(y) + g(x)g(y)
$$
for all $x, y \in \mathbb{R}$.
a. Show that for any $x, y \in \mathbb{R}$, we have $g(x + y) = f(x)g(y) + g(x)f(y)$.
b. Find all pairs $f, g$ satisfying the conditions.
$$
f(x - y) = f(x)f(y) + g(x)g(y)
$$
for all $x, y \in \mathbb{R}$.
a. Show that for any $x, y \in \mathbb{R}$, we have $g(x + y) = f(x)g(y) + g(x)f(y)$.
b. Find all pairs $f, g$ satisfying the conditions.