= Solution
At the two saddles,
$$
\frac{z_\pm^2-1}{z_\pm-t}=2z_\pm=2e^{\pm i\theta},
\qquad
f(z_\pm,t)=\frac1{z_\pm-t}
=\mp\frac{i}{\sin\theta},
$$
and
$$
|\phi''(z_\pm)|=\frac1{\sin\theta}.
$$
Applying the <method of steepest descent> with the oriented angles from part (ii), and including the prefactor $1/(2^{n+1}\pi i)$, gives two conjugate contributions. If
$$
\Phi=\left(n+\frac12\right)\theta-\frac\pi4,
$$
their sum is
$$
\frac1{2\pi}\sqrt{\frac{2\pi}{n\sin\theta}}
\left(e^{i\Phi}+e^{-i\Phi}\right).
$$
Therefore the <Debye asymptotic for Legendre polynomials> is
$$
\boxed{
P_n(\cos\theta)
\sim
\sqrt{\frac{2}{\pi n\sin\theta}}\,
\cos\left(n\theta+\frac\theta2-\frac\pi4\right)}.
$$
Thus
$$
\boxed{A(\theta)=\sqrt{\frac{2}{\pi n\sin\theta}}}.
$$
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