If x, y, z are in A.P. and tan–1x, tan–1y and tan

Engineering

Mathematics

Properties of Inverse Trigonometric Function

Question

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

x = y = z

6x = 4y = 3z

2x = 3y = 6z

6x = 3y = 2z

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2y = x + z [ x, y, z are in A.P.]

2tan^–1y = tan^–1x + tan^–1z (tan^–1x, tan^–1y, tan^–1z in A.P.)

$\Rightarrow \frac{2 y}{1 - y^{2}} = \frac{x + z}{1 - xz}$ (x + z = 2y)

$\Rightarrow$ 1 – y² = 1 – xz

$\Rightarrow$ y² = xz

$\Rightarrow$ x, y, z are in G.P. and A.P. both.

Hence x = y = z.