Chapter 11.7, Problem 9P

Prove $B(n,n)=B\left(n,\frac{1}{2}\right)/2^{2n-1}.$ Hint: In (6.4), use the identity 2 sin $\theta \cos \theta =\sin 2\theta$ and put $2\theta =\varphi .$ Use this result and ( 5.3) to derive the duplication formula for $\Gamma$ functions: $\Gamma (2n)=\frac{1}{\sqrt{\pi }}{2}^{2n-1}\Gamma (n)\Gamma \left(n+\frac{1}{2}\right)$

Check this formula for the case $n=\frac{1}{4}$ by using (5.4).