Roots of polynomials combined with Trigonometric Functions

Question

If $$ f(x) = x^2 + ax + d \cos x $$, where $a$ is an integer and $d$ is a real number, what are all possible values of the tuple $(a,d)$ such that $f(x)$ and $f(f(x))$ have the same set of real roots?

Answer 1

Hint

Let $\alpha $ a root of $f$. Then $$f(f(\alpha ))=f(0)=d,$$

If $\beta $ is a root of $f(f(x))$, then $$f(f(\beta ))=f(\beta )^2+af(\beta )+d\cos(f(\beta ))=0.$$