Question

If tan⁻¹ x + tan⁻¹ y + tan⁻¹ z = tex] \frac{\pi}{2}[/tex], then prove that xy + yz + zx = 1.

Answer 1

Please see the attachment

Answer 2

Solution :

i ) Let tan^-1 x = A

=> x = tanA

ii ) tan^-1 y = B

=> y = tanB

iii ) tan^-1 z = C

=> z = tanC

A + B + C = π/2

=> A + B = π/2 - C

=> tan( A + B ) = tan( π/2 - C )

=> [(tanA+tanB)/(1-tanAtanB)] = cotC

=> (tanA+tanB)/(1-tanAtanB) = 1/tanC

=> tanC(tanA+tanB)=1-tanAtanB

=> tanAtanC+tanBtanC=1-tanAtanB

=> tanAtanC+tanBtanC+tanAtanB = 1

=> xy + yz + zx = 1

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