Question

If (tan^-1 x)² + (cot^-1 x)² = 5π²/8, then find x.

Answer 1

(tan^-1 x)² + (cot^-1 x)² = 5π²/8

⇒ (tan^-1x + cot^-1)² – 2 tan^-1x cot^-1x = 5π²/8

⇒ 16(tan^-1 x)² – 8π(tan^-1 x) – 3π² = 0

(on solving by tan^-1 x + cot^-1 x = π/2)

⇒ 16(tan^-1x – 3π/4) (tan^-1x + π/4) = 0

⇒ tan^-1x = – π/4

⇒ x = – 1 [∵ – π/2 < tan^-1 x < π/2]