Proving Wilson's Theorem Using Gauss' Lemma

[mathsss2]

Use:
$$2\cdot4\cdot...\cdot(p-1)\equiv(2-p)(4-p)\cdot...\cdot(p-1-p)\equiv(-1)^{\frac{(p-1)}{2}}\cdot1\cdot3\cdot...\cdot(p-2)$$ mod $$p$$
and
$$(p-1)!\equiv-1$$ mod p [Wilson's Theorem]
to prove
$$1^2\cdot3^2\cdot5^2\cdot...\cdot(p-2)^2\equiv(-1)^{\frac{(p-1)}{2}}$$ mod $$p$$

Relevant equations

Gauss lemma
wilson's theorem [$$(p-1)!\equiv-1$$ mod$$p$$]

The attempt at a solution
need assistance

Thanks

 

[gabbagabbahey]

Hint:

$$2 \cdot 4 \cdot \ldots \cdot (p-1)=\frac{(p-1)!}{1 \cdot 3 \cdot \ldots \cdot (p-2)}=(p-1)! \cdot \left( \frac{1}{1 \cdot 3 \cdot \ldots \cdot (p-2)} \right)$$

 

[mathsss2]

Thanks, this problem is solved.