If T_ 0 , ~T_ 1 , ~T_ 2 T_ n represent the terms in the expansion...

From the given condition, replacing $a$ by a and $-a i$ respectively, we get $(x+a i)^{n}= \left(T_{0}-T_{2}+T_{4}-\ldots \ldots\right)+i\left( T_{1}-T_{3}+T_{5}-\ldots .\right) ...(i)$ and (x-a i)^{n}= \left( T_{0}-T_{2}+T_{4}-\ldots \ldots\right)$-i\left( T_{1}-T_{3}+T_{5}-\ldots\right)...(ii) Multiplying (ii)$ and (i) we get required result i.e., $\left(x^{2}+a^{2}\right)^{n}$=\left( T_{0}-T_{2}+T_{4}-\ldots .\right)^{2}+\left(T_{1}-T_{3}+T_{5}-\ldots\right)^{2}