From my text:
Given $\cos^n(x),$ set $x=\frac{\omega}{\sqrt{n}}$, then a local expansion yields:
$\displaystyle\cos^n(x)=e^{n\log\cos(x)}=e^{(-\frac{\omega^2}{2}+O(n^{-1} \omega^4))}$
the approximation being valid as long as $\omega = O(n^{1/4})$
I do get the approximation as indicated, but I don't understand why it's valid only if $\omega = O(n^{1/4})$
It might mean "as long as" more literally, not in the sense "only if", but in the sense of "if". The asymptotic term $n^{-1}\omega^4$ will be asymptotically smaller than the leading term $\omega^2$ for $\omega = o(n^{1/2})$.