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Solve the following inequalities: 4(tan^-1x)^2 - 8(tan^-1x) + 3 < 0

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Solve the following inequalities:

4(tan-1x)2 - 8(tan-1x) + 3 < 0

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Given, 4(tan-1x)2 - 8(tan-1x) + 3 < 0

⇒ 4a2 - 8a + 3, a = tan-1x

⇒ 4a2 - 6a - 2a = 3 < 0

⇒ 2a(2a - 3) - 1(2a - 3) < 0

⇒ (2a - 1)(2a - 3) < 0

⇒ 1/2 < a < 3/2

⇒ 1/2 < tan-1x < 3/2

⇒ tan(1/2) < x < tan(3/2)

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