Question

If tan⁻¹x + tan⁻¹y + tan⁻¹z = π, then show that `1/(xy) + 1/(yz) + 1/(zx)` = 1

Answer 1

tan⁻¹x + tan⁻¹y + tan⁻¹z = π

∴ tan⁻¹x + tan⁻¹y = π − tan⁻¹z 

∴ `tan^-1 ((x + y)/(1 - xy))` = π − tan⁻¹z 

∴ `(x + y)/(1 - xy)` = tan(π − tan⁻¹z) 

∴ `(x + y)/(1 - xy)` = −tan(tan⁻¹z) 

∴ `(x + y)/(1 - xy)` = − z

∴ x + y = −z + xyz

∴ x + y + z = xyz

∴ `1/(yz) + 1/(xz) + 1/(xy)` = 1, i.e., `1/(xy) + 1/(yz) + 1/(zx)` = 1