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Question
If tan–1 (2x) + tan–1 (3x) = `π/4`, then x = ______.
Options
–1
`1/3`
`1/6`
`3/2`
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Solution
If tan–1 (2x) + tan–1 (3x) = `π/4`, then x = `underlinebb(1/6)`.
Explanation:
tan–1 (2x) + tan–1 (3x) = `π/4`
`\implies tan^-1((2x + 3x)/(1 - 6x^2)) = π/4`
`\implies (5x)/(1 - 6x^2) = tan π/4`
`\implies` 5x = 1 – 6x2
`\implies` 6x2 + 5x – 1 = 0
`\implies` 6x2 + 6x – x – 1 = 0
`\implies` 6x(x + 1) – 1(x + 1) = 0
`\implies` (x + 1)(6x – 1) = 0
∴ x = –1, x = `1/6`
When x = `1/6`, is given equation is satisfied.
When x = –1, we get the sum of two negative angles, hence discarded.
∴ x = `1/6`
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