Prove $\Gamma\left(\frac{1}{2}\right)= \sqrt\pi$, using $\Gamma(p)\Gamma(1-p) = \frac{\pi}{\sin(\pi p)}$

Question

Prove that $\Gamma\left(\frac{1}{2}\right)= \sqrt\pi$

Using $$\Gamma(p)\Gamma(1-p) = \frac{\pi}{\sin(\pi p)}$$

Answer 1

Let $p=\frac12$, we have $$\Gamma\left(\frac12\right)\Gamma\left(\frac12\right)=\frac{\pi}{\sin\frac{\pi}{2}}=\pi$$ therefore $$\Gamma\left(\frac12\right)=\sqrt \pi$$