Finding branch cut of special function

Question

I am looking for the branch cut of this function \begin{equation} f(z) = \sqrt{3-\sin^2{z}} \end{equation} I have found the branch points of this equation as \begin{equation} z_{b1} = \frac{\pi}{2} + j\ \text{arcosh}{(\sqrt{3})} , \ z_{b2} = -\frac{\pi}{2} + j\ \text{arcosh}{(\sqrt{3})} \end{equation} From here it can be seen that $\text{arcosh}(\sqrt{3})$ gives us two possible values assuming $A$ and $-A$ ($A > 0$). Thus there are a total of 4 branch points for the original $f(z)$ \begin{equation} z_{b11} = \frac{\pi}{2} + jA , \ z_{b12} = \frac{\pi}{2} - jA, \\ z_{b21} = -\frac{\pi}{2} + jA , \ z_{b22} = -\frac{\pi}{2} - jA, \end{equation} Until here I have no idea to derive the branch cut. If you get the answer or any idea, please share it with me. Thank you all!

Answer 1

$g(z)=3-\sin^2 (z)$, let $a = arcsinh(\sqrt3)$, the zeros of $g$ are simple at $Z=\pm i a + \pi \Bbb{Z}$

$f(z)=\sqrt{g(z)}$ is analytic everywhere except at the zeros of $g$ where it has some square root branch points (ie. $f(z)(z-c)^{-1/2}$ is analytic at $c$)

Let $U\subset \Bbb{C}$ open, there is a branch of $f$ analytic on $U$ iff for all closed curve $\gamma \subset U$, $\gamma$ encloses an even number of elements of $Z$.