Derivation of formula involving Gamma function?

Another hint (bit more rigorous perhaps): The Beta Function can be written as \begin{equation} B(x,y) = \frac{x+y}{x y} \prod_{n=1}^\infty \left( 1+ \dfrac{x y}{n (x+y+n)}\right)^{-1}, \end{equation} the right hand side of the above formula can be expanded \begin{eqnarray} B(x,y) &=& \frac{x+y}{x y} \prod_{n=1}^\infty \left( 1+ \dfrac{x y}{n (x+y+n)}\right)^{-1} \\ &=& \frac{x+y}{x y} \prod_{n=1}^\infty \left( \frac{n(x+y+n)+xy}{n(x+y+n)} \right)^{-1} \\ &=& \frac{x+y}{x y} \prod_{n=1}^\infty \left( \frac{n(x+y+n)}{xy+n(x+y)+n^{2}}\right) \\ &=& \frac{x+y}{x y} \prod_{n=1}^\infty \left( \frac{n(x+y+n)}{(n+x)(n+y)}\right) \end{eqnarray} Now; The Beta function has many other forms such as \begin{equation} B(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)} \end{equation} From which a relationship between your left hand side and your right hand side could become readily found once you use properties of the gamma fuction.

Hint: Use Euler's infinite product formula for the $\Gamma$ function.