Question

Tan inverse X + tan inverse Y + tan inverse z=pie prove that X+Y+Z=xyz
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Answer 1

see explanation

Explanation:

Given

`tan^(-1)`x +
`tan^(-1)`y +
`tan^(-1)` z = π

let

`tan^(-1)`x = A ,
`tan^(-1)`y = B ,
`tan^(-1)`z = C , so

x = tanA, y = tanB , z = tanC

Substituting values

A + B + C = π ( subtract C from both sides )

A + B = π - C ( take tan of both sides )

tan(A + B) = tan(π - C) = - tanC ( expand left side using addition identity for tan )

`(tanA+tanB)/(1-tanAtanB)` = - tanC ( multiply both sides by 1 - tanAtanB )

tanA + tanB = - tanC( 1 - tanAtanB) ← distribute

tanA+ tanB = - tanC + tanAtanBtanC ( add tanC to both sides )

tanA + tanB + tanC = tanAtanBtanC , that is

x + y + z = xyz