Question

If three positive numbers x, y, z are in A.P. and also tan^–1x, tan^–1y, tan^–1z are in A.P., then

a. x = y = z

b. x ≠ y = z

c. x = y ≠ z

d. none of these

Answer 1

Correct option is a. x = y = z

Answer 2

Given 2y = x + z

Given 2tan^-1y = tan^-1x + tan^-1z   

tan(2tan^-1y) = tan(tan^-1x + tan^-1z)

  (2y)/(1 - y²)  = (x + z )/( 1- xz )

(1 - y²)   =  ( 1- xz )    [ since 2y = x + z ]

y² = xz  

above equation is valid when x = y = z