If the value of : ${\cot }^{-1} 2+{\cot }^{-1} 3$ is of the form $\frac{\pi }{a}$, then the value of $a=$

Prove that : 

${\tan }^{-1}\sqrt{\frac{x\left(x+y+z\right)}{yz}}+{\tan }^{-1}\sqrt{\frac{y\left(x+y+z\right)}{zx}}+{\tan }^{-1}\sqrt{\frac{z\left(x+y+z\right)}{xy}}=\mathrm{\pi }$.

Solve the following simultaneous equations :

${\sin }^{-1} x-{\sin }^{-1} y=\frac{\mathrm{\pi }}{3}, {\cos }^{-1} x-{\cos }^{-1} y=\frac{2\mathrm{\pi }}{3}$.