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