applications of euler's reflection formula. In this post,one of the answers (in fact the answer with more upvotes) uses euler's reflection formula $\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi\,z)}$ for the gamma function $\Gamma(z)$ to evaluate the expression $\Gamma(\frac{1}{2})$.But in the comments it is said that the formula is much more advanced than evaluating $\Gamma(\frac{1}{2})$.The question that sprang to my mind when I read that comment was "whether this comment was true in general".
Out of curiosity I'd like to know whether the comment is true in general or just a matter of opinion.Possible examples of where the reflection formula is applied except for evaluating the aforementioned expression will be appreciated.  

The most useful application of this formula is in my opinion the fact that it yields an analytic continuation of the $\Gamma$-function into the the left half of the complex plane $\{z\in\mathbb{C},\Re(z)<0\}$ , excluding the poles at $z_n=-n$ of course. This fact has wide applications, like the continuation of $\zeta(s)$ as mentioned by @Gerry Myerson in the comments.
Furthermore it simplifies some integrals nicely which are related to the Beta function: 
$$
\int_0^1 t^{z-1}(1-t)^z=B(z,1-z)=\frac{\Gamma(z)\Gamma(1-z)}{\Gamma(1)}=\frac{\pi}{\sin(\pi z )}
$$
Which i find particulary cute 
To answer the first part of your question:
I think it depends, if you already know the reflection formula then this is the easy way to prove $\Gamma(\frac{1}{2})=\sqrt{\pi}$.
If you don't, there are indeed more straightforward ways, which u can read up in the linked topic.
