D'alembert functional equation

The D'Alembert functional equation is $f(x+y)+f(x-y)=2f(x)f(y)$.
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfy the functional equation for all $x,y\in\mathbb{R}$. It's well known that $f$ is of the form $f(x)=\frac{E(x)+E^∗(x)}{2}$, for some $E:\mathbb{R}\rightarrow\mathbb{C}$.
How can I use this functional equation to solve the following problem?

Let $\lambda$ be a nonzero real constant. Find all functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy the functional equation $f(x+y)+g(x-y)=\lambda f(x)g(y)$
for all $x,y\in\mathbb{R}$.