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

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

The correct answer is Option (2) → $[-\sqrt{5},-\sqrt{3}] ∪ [\sqrt{3},\sqrt{5}]$

For $y=\cos^{-1}(x^{2}-4)$ the expression inside $\cos^{-1}$ must lie in the interval:

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

Add 4 to all sides:

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

So:

$x^{2} \ge 3$ and $x^{2} \le 5$

Taking square roots:

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

Hence the domain is:

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

The domain is $[-\sqrt{5}, -\sqrt{3}] \cup [\sqrt{3}, \sqrt{5}].$