Question

If tan^–1x + tan^–1y + tan^–1z = π, show that x + y + z = xyz

Answer 1

tan^–1x + tan^–1y + tan^–1z = π

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

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

`tan^-1 [((x + y + z(1 - xy))/(1 - xy))/((1 - xy - (xz + yz))/(1 - xy))] = pi`

`(x + y + z - zyz)/(1 - xy - xz - yz) = tanpi` = 0

x + y + z – xyz = 0

x + y + z = xyz