How to prove $\frac{\pi}{4 \Gamma^2(1.5)} =1$?

Question

I obtained the following identity, by numerical analysis. $\Gamma$ is the Gamma function. How can I analytically show this?

\begin{equation} \frac{\pi}{4 \Gamma^2(1.5)} =1 \end{equation}

Answer 1

See here for proof that $\Gamma(1/2) = \sqrt{\pi}$.

Then apply that $\Gamma(z+1)=z \Gamma(z)$.