Given a real value $-1 \leq m \leq 1$, I need to determine the asymptotic $\left(~\mbox{large}\ N~\right)$ behavior of the following integral:
$$ \int_{- \pi}^{\pi} \frac{\mathrm{d}\hat{m}}{2\pi}\, \left[\,{\mathrm{e}^{\mathrm{i}\,\hat{m}\,m}\, \cos\left(\hat{m}\right)}\,\right]^{N} $$
It should satisfy some large deviation principle. How can I derive it $?$.
$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\on}[1]{\operatorname{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$ $\ds{\bbox[5px,#ffd]{}}$
\begin{align} &\bbox[5px,#ffd]{% \left.\int_{- \pi}^{\pi} {\dd\theta \over 2\pi}\, \bracks{\expo{\ic m\theta}\, \cos\pars{\theta}}^{N} \,\right\vert_{\ m\ \in\ \bracks{-1,1}}} \\[5mm] = &\ {1 \over \pi} \int_{0}^{\pi}\cos\pars{Nm\theta} \cos^{N}\pars{\theta}\,\dd \theta \\[5mm] = &\ {\pars{-1}^{N} \over \pi} \int_{-\pi/2}^{\pi/2}\cos\pars{Nm\theta + Nm\,{\pi \over 2}} \sin^{N}\pars{\theta}\,\dd \theta \\[5mm] = &\ {\pars{-1}^{N} \over \pi} \\[2mm] &\ \int_{0}^{\pi/2}\left[% \cos\pars{Nm\theta + Nm\,{\pi \over 2}} \sin^{N}\pars{\theta}\right. \\[2mm] &\ \left. \cos\pars{-Nm\theta + Nm\,{\pi \over 2}} \sin^{N}\pars{-\theta} \right]\dd \theta \\[5mm] = &\ {\pars{-1}^{N} \over \pi} \int_{0}^{\pi/2}\left[% \cos\pars{Nm\pi - Nm\theta} \cos^{N}\pars{\theta}\right. \\[2mm] &\ \phantom{{\pars{-1}^{N} \over \pi} \int_{0}^{\pi/2}\left[\,\,\right.} \left. \cos\pars{Nm\theta}\pars{-1}^{N} \cos^{N}\pars{\theta} \right]\dd \theta \\[5mm] = &\ {\pars{-1}^{N} \over \pi} \\[2mm] &\ \int_{0}^{\pi/2} \bracks{\cos\pars{Nm\theta - Nm\pi} + \pars{-1}^{N}\cos\pars{Nm\theta}} \\[2mm] &\ \expo{N\ln\pars{\cos\pars{\theta}}} \,\,\,\dd\theta \\[5mm] &\ \stackrel{\color{red}{\mrm{as}\ N\ \to\ \infty}}{\sim}\,\,\, {\pars{-1}^{N} \over \pi} \bracks{\cos\pars{Nm\pi} + \pars{-1}^{N}} \\[2mm] &\ \int_{0}^{\infty} \exp\pars{-N\theta^{2} \over 2}\dd\theta \\[5mm] = &\ {1 \over \root{2\pi}} {1 + \pars{-1}^{N}\cos\pars{Nm\pi} \over N^{1/2}} \end{align}