How to proof with the definition of gamma function?

Question

$\Gamma(\dfrac{1}{2}-z)\Gamma(\dfrac{1}{2}+z) = \dfrac{\pi}{cos \pi z}$

thanks for the comment this problem has been solved

Answer 1

Use the well-known Euler's reflection formula, namely,

\begin{equation} \Gamma{(z)}\Gamma{(1 - z)} = \dfrac{\pi}{\sin(\pi z)}. \end{equation}

Taking $z = \frac{1}{2} - z$ yields the equation that you are looking for.

The proof for Euler's reflective equation is given here https://proofwiki.org/wiki/Euler%27s_Reflection_Formula