Binomial/combinatorics summation proof question

Given $sum_{k=0}^{2r} (-1)^k binom{n}{k}binom{n}{2r-k} = (-1)^rbinom{n}{r}$ for $0≤r≤frac12n$

How do I show that $sum_{k=0}^{r} (-1)^k binom{n}{k}binom{n}{2r-k} = frac12(-1)^rbinom{n}{r}(1+binom{n}{r})$ for $0≤r≤frac12n$

For context, the first statement is derived from considering the coefficient of $x^{2r}$ in the statement $(1-x)^n(1+x)^n=(1-x^2)^n$ where $2r≤n$