Prove $cos(\sqrt{z})-z$, $z\in \mathbb C$ has infinitely many zeroes (without Picard's theorem or more advanced results)

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Found this question on an exam online and have been stuck on it for awhile. I'm knowledgeable up to roughly chapter 9 section 1 of Greene and Krantz Function Theory of one complex variable 3rd edition (I have taken a first graduate level course in CA). I have seen other solutions on here but they involve more advanced results like Picard's theorem. If possible I'd like a solution which only requires the knowledge I have currently.

I also have some basic knowledge of SCV (roughly up to the end of Chapter 2 of R Michael Range's Holo. functions and integral rep. in several complex variables, i.e. I know basic facts about holomorphic functions in several variables, domains of holomorphy, pseudoconvexity) so a solution using basic results from several variables would also be fine.

I have also heard Jensen's formula can be used to prove it, but I could not figure out how to apply it.