From $\cos(x-y)$ to $f(x+y)$?

Question

Is it possible to transform $\cos(x-y)$ into a function $f=f(x+y)$ to have: $$\cos(x-y)=f(x+y)$$?

Answer 1

If it is true for $x=y$ you have that

$1=f(2x)$ so

$f(t)=f(2(\frac{t}{2}))=1$

But it is not possibile because $f=1$ not verify the condition $cos(x-y)=f(x-y)$.

So there is not a function that verify your condition

Answer 2

No, because $\cos(1-1)$ is different from $\cos(2-0)$, so $f(2)$ can't be both of them at once.