Mathematics - Binomial Theorem Question with Solution | TestHub

Put $\mathrm{x}=\mathrm{i}$ in the expansion of $(1+x{)}^n$ we get $\begin{array}{rlr}& \left(C_0-C_2+C_4-C_6\dots \dots \right)+i\left(C_1-C_3+C_5\dots .\right)=2^{n/2}\left[\cos \frac{n\pi }{4}+i\sin \frac{n\pi }{4}\right]\text{ Take modulus both sides and square it }& \Rightarrow {\left(C_0-C_2+C_4-C_6+\dots \right)}^2+{\left(C_1-C_3+C_5-C_7+\dots \right)}^2=2^n\end{array}$