If `x + y + z = xyz and x, y, z gt 0`, then find the value of `tan^(-1) x + tan^(-1) y + tan^(-1) z`

Answer

`pi`

Explanation

`xy = 1 _ (x)/(z) + (y)/(z) gt 1`
`rArr tan^(-1) x + tan^(-1) y + tan^(-1) z`
`= pi + tan^(-1) [(x + y)/(1 - xy)] + tan^-(1) z`
`= pi + tan^(-1) [(xyz -z)/(1 - xy)] + tan^(-1) z`
`= pi + tan^(-1) [(z (xy -1))/(1 -xy)] + tan^(-1) z`
`= pi + tan^(-1) (-z) + tan^(-1) z = pi`