In a poisson distribution, the variance is m. The sum of terms in odd places in the distribution is

Sum of Odd terms $=P\left(X=0\right)+P\left(X=2\right)+P\left(X=4\right)+\mathrm{...........}$

                                   $=e^{-m}\left[1+\frac{m^2}{2!}+\frac{m^4}{4!}+\mathrm{..........}\right]$

     $=\frac{e^{-m}}{2}\left[\left[1+\frac{m}{1!}+\frac{m^2}{2!}+\frac{m^3}{3!}+\mathrm{...........}\right]+\left[1-\frac{m}{1!}+\frac{m^2}{2!}-\frac{m^3}{3!}+\mathrm{...........}\right]\right]$

$=e^{-m}\left[\frac{e^m+e^{-m}}{2}\right]=e^{-m}\cosh \left(m\right)$