The domain of $y=cos^{-1}(x^2-4)$ is :

$[-\sqrt{5}, -\sqrt{3}]∪[\sqrt{3}, \sqrt{5}]$

Given function: $y=\cos^{-1}(x^{2}-4)$

Domain condition: Argument of $\cos^{-1}$ must lie in $[-1,1]$.

$-1 \leq x^{2}-4 \leq 1$

Add $4$ to all sides:

$3 \leq x^{2} \leq 5$

Taking square root:

$\sqrt{3} \leq |x| \leq \sqrt{5}$

So, $x \in [-\sqrt{5},-\sqrt{3}] \cup [\sqrt{3},\sqrt{5}]$

Domain = $[-\sqrt{5},-\sqrt{3}] \cup [\sqrt{3},\sqrt{5}]$