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Question
If tan-12x + tan-13x = `pi/4`, then find the value of x.
Solution
tan-12x + tan-13x = `pi/4`
∴ `tan^-1 ((2"x" + 3"x")/(1 - "2x" xx "3x")) = pi/4`, where 2x > 0, 3x > 0
∴ `"5x"/(1 - "6x"^2) = tan pi/4 = 1`
∴ 5x = 1 - 6x2
∴ 6x2 + 5x - 1 = 0
∴ 6x2 + 6x - x - 1 = 0
∴ 6x(x + 1) - 1(x + 1) = 0
∴ (x + 1)(6x - 1) = 0
∴ x = -1 or x = `1/6`
But x > 0 ∴ x ≠ - 1
Hence, x = `1/6`
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