Question

Prove that:(its urgent plz help someone who's intelligent enough !!)
$tan^-^1 \frac{x-y}{1+xy} +tan^-^1 \frac{y-z}{1+yz} +tan^-^1 \frac{z-x}{1+zx} =tan^-^1 \frac{ x^{3}- y^{3} }{1+ x^{3} y^{3} }+tan^-^1 \frac{ y^{3}- z^{3} }{1+ y^{3} z^{3} }$$+tan^-^1 \frac{ z^{3}- x^{3} }{1+ z^{3} x^{3} }$

Answer 1

Let x = tan A    ,     y = Tan B      ,   z = Tan C

(x - y)/(1+x y) = tan (A-B)
(y-z)//(1+y z) = tan (B-C)
(z-x)/(1+z x) = tan (C - A)

LHS = (A-B) + (B-C) + (C-A) = 0

RHS is also similar to LHS.
If we assume x³ = tan P ,    y³ = tan Q   ,   z³ = tan R
RHS = tan⁻¹ (tan (P-Q) ) + tan⁻¹ [tan (Q-R) ] + tan⁻¹ [ tan (R-P) ] = 0

LHS = RHS