If n is odd, then Coefficient of xⁿ in ${\left(1+\frac{{\mathrm{x}}^2}{2!}+\frac{{\mathrm{x}}^4}{4!}+\dots \dots \infty \right)}^2=$

 

$\begin{array}{rcl} & & {\left(1+\frac{{\mathrm{x}}^2}{\overline{)\underline{2}}}+\frac{{\mathrm{x}}^4}{\overline{)\underline{4}}}+\dots \dots \infty \right)}^2\\ & =& {\left(\frac{{\mathrm{e}}^{\mathrm{x}}+{\mathrm{e}}^{-\mathrm{x}}}{2}\right)}^2\\ & =& \frac{1}{4}\left({\mathrm{e}}^{\mathrm{ex}}+{\mathrm{e}}^{-2\mathrm{x}}+2\right)\\ & =& \frac{1}{2}\left({\mathrm{e}}^{2\mathrm{x}}+{\mathrm{e}}^{-2\mathrm{x}}+1\right)\\ & =& \frac{1}{2}\left(\cos \mathrm{h} 2\mathrm{x}+1\right)\\ & =& \frac{1}{2}\left[\left(1+\frac{{\left(2\mathrm{x}\right)}^2}{\overline{)\underline{4}}}+...\right)+1\right]\\ & & \end{array}$
$\begin{array}{rcl}if n is odd ,x^n coef is & =& 0\end{array}$