Magnitude of $\Gamma(1/2+it)$

Question

I want to prove that $$|\Gamma(1/2+it)|=\sqrt{\frac{2\pi}{e^{\pi t}+e^{-\pi t}}}$$ My idea is to use the formula $\Gamma(s)\Gamma(1-s)=\pi/\sin \pi s$. Plugging in $s=1/2+it$ and taking absolute values, we get $$|\Gamma(1/2+it)\Gamma(1/2-it)|=\dfrac{\pi}{\sin(\frac{\pi}{2}+it\pi)}$$ How to continue from here?

Answer 1

We have $$\Gamma(\bar{z})=\overline{\Gamma(z)}$$ as well as $$| \sin z|^2=\cosh^2 y-\cos^2 x. $$

Answer 2

Well, is it true that $|\Gamma(x)|$ is conjugation invariant on the line $\Re x= \frac12?$ If yes, than what is $\sin (\pi/2 + z)?$ When you do that, substitute $z = it \pi,$ and use the exponential expression for $\sin.$