Prove that $\Gamma(-k+\frac{1}{2})=\frac{(-1)^k 2^k}{1\cdot 3\cdot 5\cdots(2k-1)}\sqrt{\pi}$. I was able to prove that
$$
\Gamma\left (k+\frac{1}{2} \right )=\frac{1\cdot 3\cdot 5\cdots(2k-1)}{2^k}\sqrt{\pi}.\tag{$k\geq 1$}$$
using the Legendre's duplication formula. But I can't do the same to 
$$\Gamma\left ( -k+\frac{1}{2} \right )=\frac{(-1)^k 2^k}{1\cdot 3\cdot 5\cdots(2k-1)}\sqrt{\pi}.\tag{$k\geq 1$}$$
If possible, I'd like you to give a hint. If it is not possible to use Legendre's duplication formula, then I tried this way, for $n\geq 1$,
\begin{align*}
\Gamma\left ( -n+\frac{1}{2} \right )&=\left ( -n-\frac{1}{2} \right )\Gamma\left ( -n-\frac{1}{2} \right )\\
&=\dots\\
&=\left ( \frac{1}{2} \right )\left ( \frac{3}{2} \right )\cdots\left ( \frac{-2n-3}{2} \right )\left ( \frac{-2n-1}{2} \right )\Gamma\left ( \frac{1}{2} \right )
\end{align*}
and not anymore. How do I count how many factors there are to right side beside $\Gamma(1/2)$?

If you are willing to accept that for $z \notin \Bbb Z$
$$\Gamma (z) \Gamma (1-z) = \frac \pi {\sin \pi z}$$
then letting $z = \frac 1 2 - n$ leads to
$$\Gamma \left( \frac 1 2 - n \right) \Gamma \left( \frac 1 2 + n \right) = \frac \pi {\sin \left( \frac \pi 2 - n \pi \right)} = \frac \pi {\cos n \pi} = (-1)^n \pi$$
and since you already know
$$\Gamma \left( n + \frac 1 2 \right) = \frac {1 \cdot 3 \cdot 5 \cdot \cdots \cdot (2n-1)} {2^n} \sqrt{\pi}$$
then
$$\Gamma \left( \frac 1 2 - n \right) = \frac {(-1)^n 2^n \sqrt \pi} {1 \cdot 3 \cdot 5 \cdot \cdots \cdot (2n-1)} .$$
Alternatively, your own attempt based upon $\Gamma (z) = (z-1) \Gamma (z-1)$ is fine, too, because
$$\Gamma \left( \frac 1 2 \right) = \left( \frac 1 2 - 1 \right) \Gamma \left( \frac 1 2 - 1 \right) = \\
\left( \frac 1 2 -1 \right) \left( \frac 1 2 - 2 \right) \Gamma \left( \frac 1 2 - 2 \right) = \dots \\
\underbrace { \left( \frac 1 2 -1 \right) \left( \frac 1 2 - 2 \right) \dots \left( \frac 1 2 - n \right) } _{n \text{ factors}} \Gamma \left( \frac 1 2 - n \right) = \\
\underbrace { \left( - \frac 1 2 \right) \left( - \frac 3 2 \right) \dots \left( - \frac  {2n-1} 2 \right) } _{n \text{ factors}} \Gamma \left( \frac 1 2 - n \right) = (-1)^n \frac {1 \cdot 3 \cdot \dots \cdot (2n-1)} {2^n} \Gamma \left( \frac 1 2 - n \right)$$
whence the desired result follows.
