If $C_0,C_1,C_2,\dots ,C_n$ are coefficients in the expansion of $(1+x{)}^n$ and $n$ is even , then

$C_0^2-C_1^2+C_2^2-C_3^2+\dots +(-1{)}^n{C}_n^2$ is equal to 

If n is even, suppose $(r+1{)}^{\text{th }}$ term in the binomial expansion of ${\left(1-x^2\right)}^n$ contains $x^n$

We have,

$T_{r+1}{=}^n{C}_r(-1{)}^r{\left(x^2\right)}^r{=}^n{C}_r(-1{)}^r{x}^{2r}$

For this term to contain $x^n$, we must have

$2r=n\Rightarrow r=n/2$

Coefficient of $x^n{=}^n{C}_{n/2}(-1{)}^{n/2}$

Also, Coefficient of $x^n\text{ in }{\left(1-x^2\right)}^n$ (see illustration 18) is

$\begin{array}{l}{C}_0^2-C_1^2+C_2^2-C_3^2+\dots +(-1{)}^n{C}_n^2\\ C_0^2-C_1^2+C_2^2-C_3^2+\dots +(-1{)}^n{C}_n^2{=}^n{C}_{n/2}(-1{)}^{n/2}\end{array}$