Find all entire functions $f$ for which there exists a positive constant $M$ such that $|f(z)|\le M|\cos z|$ for all $z\in \mathbb{C}$.
I think to prove this, we need to use the Identity theorem, i.e. analytic functions are determined by their values in a convergent sequence. In this case, we can find real sequence converging to $\pi/2$. In that case, given any $\epsilon >0$, eventually $|f(z_n)|$ must be less than $M\epsilon$. But this does not guarantee that $f(z_n)=0$, which is what I need to prove that $f\equiv 0$. How can I sharpen this proof? I would greatly appreciate any help.
Wrong theorem ,use Liouville's Theorem -|f(z)/cos(z)| <= M (note the singularities when cos(z) =0 are removable so the quotient is a bounded entire function haence must be constant which gives F(z)=A cos(z) with A a fixed complex number