Maharashtra State BoardHSC Arts 12th Board Exam
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If tan−1x + tan−1y + tan−1z = π, then show that 1xy+1yz+1zx = 1 - Mathematics and Statistics

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Sum

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

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Solution

tan−1x + tan−1y + tan−1z = π

∴ tan−1x + tan−1y = π − tan−1

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

∴ `(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

Concept: Inverse Trigonometric Functions
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