Why is $\cos \sqrt z$ entire but $\sin \sqrt{z}$ isn't?

Hint Suppose $F(z) := f(\sqrt{z})$ is analytic at $z = 0$ for some choice of branch cut of $\sqrt{\cdot}$, say, $F(w)$ has power series $$F(w) \sim a_0 + a_1 w + a_2 w^2 + \cdots$$ at $w = 0$. What is the power series of $f(w) = F(w^2)$ at $w = 0$?

Additional hint We have $$f(w) \sim a_0 + a_1 w^2 + a_2 w^4 + \cdots .$$

This shows that not only is $\sin \sqrt{z}$ not entire, it's not analytic in any neighborhood of $z = 0$ (for any choice of branch cut).