Question

If tan^–1 2x + tan^–1 3x = `π/4`, then x = ______.

Options

• –1

• `1/3`

• `1/6`

• `1/2`

Answer 1

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`