If $\mathrm{x}, \mathrm{y}, \mathrm{z}$ are in A.P. and $\tan ^{-1} \mathrm{x}, \tan ^{-1} \mathrm{y}$ and $\tan ^{-1} \mathrm{z}$ are also in A.P., then

JEE Main · 2013 · 07 Apr 2013 · Offline · Inverse Trigonometric Functions

x, y, z

are in A.P. and

tan^{- 1} x, tan^{- 1} y

and

tan^{- 1} z

are also in A.P., then

Answer: (D)

Being in A.P. means

2 y = x + z

, and the inverse tangents being in A.P. means

2 tan^{- 1} y = tan^{- 1} x + tan^{- 1} z

. Taking the tangent of both sides,

\frac{2 y}{1 - y ^{2}} = \frac{x + z}{1 - x z}

. Since

x + z = 2 y

, the numerators agree, forcing

1 - y^{2} = 1 - x z

, so

y^{2} = x z

. Together with

2 y = x + z

this is the equality case of the A.M.-G.M. relation, which holds only when

x = y = z

JEE Main · 2013 · 07 Apr 2013 · Offline · Single correct option

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