The original problem is to prove that if p $\equiv 1 \mod{4}$, then $[((p-1)/2)!]^2 \equiv -1 \mod{p}$, and the problem in the title is given as a hint.
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Use the fact that $-a \equiv p - a \pmod{p}$ many times, and leverage Wilson's theorem. – Jakob Streipel Nov 27 '21 at 17:10
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See also, from some brief searching, e.g. https://math.stackexchange.com/q/1485760. It's for the $p \equiv 3 \pmod{4}$ case, but it's a minor detail toward the end only. – Jakob Streipel Nov 27 '21 at 17:18