Integration of Legendre Polynomials with different arguments I'm trying to calculate this: 
$$\sum_{l=0}^{\infty} \int_{\Omega} d\theta' d\phi' \cos{\theta'} \sin{\theta'} P_l (\cos{\gamma})$$
where $P_{l}$ are the Legendre polynomials, $\Omega$ is the surface of a sphere of radius $R$, and
$$ \cos{\gamma} = \cos{\theta'} \cos{\theta} + \sin{\theta'}\sin{\theta}\,\cos({\phi' -\phi}) $$ 
I am told that only the $l=1$ term survives due to orthogonality of Legendre polynomials (of course $\cos{\theta'} = P_{1}(\cos{\theta'}) $), but I'm don't see why, since the Legendre polynomials have different arguments. 
How can I show that this is true?
 A: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
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 \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
 \newcommand{\ic}{\mathrm{i}}
 \newcommand{\mc}[1]{\mathcal{#1}}
 \newcommand{\mrm}[1]{\mathrm{#1}}
 \newcommand{\pars}[1]{\left(\,{#1}\,\right)}
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\begin{align}
&\sum_{\ell = 0}^{\infty}\int_{\Omega'}\dd\theta'\,d\phi'\,\cos{\theta'} \sin{\theta'}\,\mrm{P}_{\ell}\pars{\cos\pars{\gamma}}
\\[5mm] = &\
\sum_{\ell = 0}^{\infty}\int_{\Omega'}
\overbrace{\bracks{%
2\root{\pi \over 3}\,\mrm{Y}_{10}\pars{\Omega'}}}^{\ds{\cos\pars{\theta'}}}\
\overbrace{\bracks{{4\pi \over 2\ell + 1}\sum_{m = -\ell}^{\ell}
\mrm{Y}_{\ell m}\pars{\Omega}\,\mrm{Y}_{\ell m}^{*}\pars{\Omega'}}}
^{\ds{\mrm{P}_{\ell}\pars{\cos\pars{\gamma}}}}\
\dd\Omega'
\\[5mm] = &\
4\pi
\sum_{\ell = 0}^{\infty}\sum_{m = -\ell}^{\ell}
2\root{\pi \over 3}{\mrm{Y}_{\ell m}\pars{\Omega}\, \over 2\ell + 1}\
\underbrace{\int_{\Omega'}
\mrm{Y}_{10}\pars{\Omega'}\,\mrm{Y}_{\ell m}^{*}\pars{\Omega'}\dd\Omega'}
_{\ds{\delta_{\ell 1}\,\delta_{m0}}}
\\[5mm] = &\
4\pi\bracks{{1 \over 2 \times 1 + 1}
\,2\root{\pi \over 3}\mrm{Y}_{10}\pars{\Omega}} =
\bbx{{4\pi \over 3}\,\cos\pars{\theta}}\quad\mbox{because}\quad
2\root{\pi \over 3}\,\mrm{Y}_{10}\pars{\Omega} = \cos\pars{\theta}
\end{align}
