Solve: tan-1(x + 1) + tan-1(x - 1) = tan-1(8/31)

1.	Solve: tan-1(x + 1) + tan-1(x - 1) = tan-1(8/31)
Answer» tan^-1(x + 1) + tan^-1(x - 1) = tan^-1(8/31) ⇒ tan^-1(((x + 1) + (x - 1))/(1 - (x + 1).(x - 1))) = tan^-1(8/31) ⇒ tan^-1(2x/(1 - (x² - 1))) = tan^-1(8/31) ⇒ tan^-1(2x/(2 - x²)) = tan^-1(8/31) ⇒ 2x/(2 - x²) = 8/31 ⇒ 62x = 16 - 8x² ⇒ 8x² + 62x - 16 = 0 ⇒ 4x² + 31x - 8 ⇒ (4x - 1)(x + 8) = 0 ⇒ x = 1/4 and x = - 8 As x = - 8 does not satisfy the equation Hence x = 1/4 is only solution.

Answer»

tan^-1(x + 1) + tan^-1(x - 1) = tan^-1(8/31)

⇒ tan^-1(((x + 1) + (x - 1))/(1 - (x + 1).(x - 1))) = tan^-1(8/31)

⇒ tan^-1(2x/(1 - (x² - 1))) = tan^-1(8/31)

⇒ tan^-1(2x/(2 - x²)) = tan^-1(8/31)

⇒ 2x/(2 - x²) = 8/31 ⇒ 62x = 16 - 8x² ⇒ 8x² + 62x - 16 = 0

⇒ 4x² + 31x - 8 ⇒ (4x - 1)(x + 8) = 0

⇒ x = 1/4 and x = - 8

As x = - 8 does not satisfy the equation

Hence x = 1/4 is only solution.

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