If the sum of odd numbered terms and the sum of even numbered terms in the expansion of ${(x+a)}^n$  are  A and B respectively, then the value of ${(x^2-a^2)}^n$  is

Given expansion ${(x+a)}^n$  then we know that its expansions

${(x+a)}^n={}^n{\text{C}}_0{x}^n+{}^n{\text{C}}_1{x}^{n-1}a$   ${+}^n{\text{C}}_2{x}^{n-2}{a}^2+\mathrm{....}+{}^n{\text{C}}_n{a}^n$        … (1)

$\Rightarrow {(x+a)}^n=({}^n{\text{C}}_0{x}^n+{}^n{\text{C}}_2{x}^{n-2}{a}^2+\mathrm{...})$ $+({}^n{\text{C}}_1{x}^{n-1}a+{}^n{\text{C}}_3{x}^{n-3}{a}^3+\mathrm{...})$               … (2)

$A=({}^n{\text{C}}_0{x}^n+{}^n{\text{C}}_2{x}^{n-2}{a}^2+\mathrm{...})$ (odd number terms),$B=({}^n{\text{C}}_1{x}^{n-1}a+{}^n{\text{C}}_3{x}^{n-3}{a}^3+\mathrm{...})$ (even number terms)

$\Rightarrow {(x+a)}^n=A+B\mathrm{..}.\left(2\right)$ , where

              $A={}^n{\text{C}}_0{x}^n+{}^n{\text{C}}_2{x}^{n-2}{a}^2+\mathrm{...}$

And   $B={}^n{\text{C}}_1{x}^{n-1}a+{}^n{\text{C}}_3{x}^{n-3}{a}^3+\mathrm{...}$

Changing a to $-a$  in ${(x+a)}^n$ Eq (i) we get

${(x-a)}^n=A-B$                                                …(3)   

Multiplying Eq(2) and Eq(3), we get

${(x+a)}^n{(x-a)}^n=\left(A+B\right)\left(A-B\right)$

     ${(x^2-a^2)}^n=A^2-B^2$ .