Analytic Branches

Question

How would I show that there exists an analytic branch of $$(1-cosz^2)^\frac{1}{4}$$ near $z=0$? My initial thoughts is to use power series expansion since, the expansion of $cos z $ converges for all $z$. This gives $$(1-cosz^2)^\frac{1}{4} = 2^{\frac{-1}{4}}z + 24^{\frac{-1}{4}}z^2+...$$ But now how do i show the existence of an analytic branch?