Find a function f : R → R such that: f(x + y) − f(x − y) = f(x)f(y), ∀x, y ∈ R.

To find a function f: \mathbb{R} \rightarrow \mathbb{R} that satisfies the given functional equation:

f(x + y) - f(x - y) = f(x)f(y), \forall x, y \in \mathbb{R}

1. Start by setting y = 0 in the functional equation:
f(x + 0) - f(x - 0) = f(x)f(0)
f(x) - f(x) = f(x)f(0)
0 = f(x)f(0)

This implies that for all x , we have f(x)f(0) = 0 . If f(0) \neq 0 , then f(x) = 0 for all x , leading to a contradiction. Thus, we conclude f(0) = 0 .

2. Substituting x = 0 into the original equation:
f(y) - f(-y) = f(0)f(y)
f(y) - f(-y) = 0
f(y) = f(-y)

This shows that f is an even function.

3. A simple non-trivial solution is f(x) = 0 for all x .
Plugging into the original equation:
f(x + y) - f(x - y) = 0
f(x)f(y) = 0

Therefore, f(x) = 0 is a valid non-trivial solution satisfying the functional equation.

\boxed{f(x) = 0} is a non-trivial solution to the given functional equation.