If `x > y > z >0,` then find the value of `cot^(-1)(x y+1)/(x-y)+cot^(-1)(y z+1)/(z y-z)+cot^(-1)(z x+1)/(z-x)`

Explanation

`cot^(-1).(xy + 1)/(x -y) + cot^(-1).(yz + 1)/(y - z) + cot^(-1).(zx + 1)/(z - x)`
`tan^(-1).(x -y)/(1 + xy) + tan^(-1).(y -z)/(1 + yz) + pi + tan^(-1).(z - x)/(1 + zx)`
`= [ :' tan^(-1) ((1)/(x))= {(cot^(-1) x," for " x gt0),(-pi + cot^(-1) x,"for " x lt 0):}]`
`= tan^(-1) x - tan^(-1) y + tan^(-1) y - tan^(-1) z + pi + tan^(-1) z - tan^(-1) x`
`= pi`