This functional equation is known as D'Alembert's functional equation. To solve it, we can make the substitution:

a = x + y
b = x - y

which gives us:

x = (a + b)/2
y = (a - b)/2

Now, we substitute these values into the given functional equation:

f(a) + f(b) = 2f((a + b)/2)cos((a - b)/2)

This equation is known as D'Alembert's functional equation. It is a well-known functional equation in mathematics.