Suppose that $X_1,\ldots,X_n$ form a random sample from the Bernoulli distribution with paremeter $p$. Let $\bar{X}_n$ be the sample average. Use the variance stabilizing transformation $\alpha(x)=\arcsin(\sqrt{x})$ to construct an approximate coefficient $\gamma$ confidence interval for $p$.
The solution says the variance stabilizing function is $\alpha(x)=\arcsin(\sqrt{x})$, and the approximate distribution of $\alpha(\bar{X}_n)$ is the normal distribution with mean $\alpha(p)$ and variance $1/n$. So,
$$\mathbb{P}\left(\arcsin(\sqrt{\bar{X}_n})-\Phi^{-1}([1+\gamma]/2)\sqrt{n}<\arcsin(\sqrt{p})<\arcsin(\sqrt{\bar{X}_n})+\Phi^{-1}([1+\gamma]/2)\sqrt{n}\right)\approx\gamma.$$
Then the solution continues, but I don't get the start of it.