Prove that `(""^(2n)C_(0))^2-(""^(2n)C_(1))^2+(""^(2n)C_(2))^2-.....+(-1)^n(""^(2n)C_(2n))^2=(-1)^n.""^(2n)C_(n)`

Explanation

To prove the identity

\[ \left( \binom{2n}{0} \right)^2 - \left( \binom{2n}{1} \right)^2 + \left( \binom{2n}{2} \right)^2 - \ldots + (-1)^n \left( \binom{2n}{2n} \right)^2 = (-1)^n \binom{2n}{n}, \]

we will use the properties of binomial coefficients and the binomial theorem.

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