According to the so-called formula of Laplace (see, e.g., [1, Theorem 8.21.2]), Legendre polynomials admit the following asymptotic expansion: $$ P_n(\cos\theta) = \sqrt{2} (\pi n\sin\theta)^{-\frac12}\cos((n+\tfrac12)\theta - \tfrac{\pi}{4}) + O(n^{-\frac32}), \quad \text{as } \; n \to +\infty, $$ where the convergence is uniform for $\theta \in [\varepsilon,\pi-\varepsilon]$.

Question: I would like to understand the dependence of the error term on $\varepsilon$, but I haven't found anything useful in this direction in the literature so far. Do you know where could I find this kind of result (if available)?

Bibliography

[1] Szegö, G. Orthogonal polynomials, 1939.