Question

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

Answer 1

Answer:

see explanation

Step-by-step 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 )

$\frac{tanA+tanB}{1-tanAtanB}$ = - 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