If x , y , z are in arithmetic progression and tan - 1 ⁡ x , tan - 1 ⁡ y and tan - 1 ⁡ z are also in arithmetic progression, then

Question

If x , y , z are in arithmetic progression and tan - 1 ⁡ x , tan - 1 ⁡ y and tan - 1 ⁡ z are also in arithmetic progression, then, x = y = z, x = y = - z, x = 1 , y = 2 , z = 3, x = 2 , y = 4 , z = 6

MARKS App · Accepted Answer

Since, x , y , z are in AP ∴ y = x + z 2 ....(i) And tan - 1 ⁡ x , tan - 1 ⁡ y and tan - 1 ⁡ z are also in AP. ∴ 2 tan - 1 ⁡ y = tan - 1 ⁡ x + tan - 1 ⁡ z ⇒ tan - 1 ⁡ 2 y 1 - y 2 = tan - 1 ⁡ x + z 1 - x z ⇒ 2 y 1 - y 2 = 2 y 1 - x z [from equation. (i)] ⇒ y 2 = x z ⇒ x , y , z are in GP. ∴ x = y = z

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