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would any one tell me: which of the function is entire $\sin(\sqrt{z})$ and $\cos(\sqrt{z})$. I have no Idea in which logic I need to apply here. Thank you.

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Have you tried looking at the Maclaurin series of these two functions? – J. M. Aug 4 '12 at 17:06
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Morally: the "function" $z \mapsto \sqrt{z}$ fails to be entire because of the ambiguity in the square root: for any $z$ there are two choices of square root, and there's no continuous way to make a consistent choice on the whole complex plane. When you compute $\sin(\sqrt{z})$, does it depend on which choice of square root you made? What about $\cos(\sqrt{z})$? – countinghaus Aug 4 '12 at 17:12
Thank you....... – miosaki Aug 4 '12 at 19:51

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