Question

tan⁻¹2x + tan⁻¹3x= π/4,Solve it.

Answer 1

it is given that, tan⁻¹2x + tan⁻¹3x = π/4

we know,

tan⁻¹A + tan⁻¹B = tan⁻¹[(A + B)/(1 - AB)], AB < 1

so, tan⁻¹(2x) + tan⁻¹(3x) = tan⁻¹[(2x + 3x)/(1 - 2x × 3x)] , (2x)(3x)< 1 ⇒x² < 1/6

= tan⁻¹[5x/(1 - 6x²)]

or, tan⁻¹[5x/(1 - 6x²)] = π/4

or, 5x/(1 - 6x²) = tan(π/4)

or, 5x/(1 - 6x²) = 1

or, 5x = 1 - 6x²

or, 6x² + 5x - 1 = 0

or, 6x² + 6x - x - 1 = 0

or, 6x(x + 1) - 1(x + 1) = 0

or, (6x -1)(x + 1) = 0 ⇒x = 1/6, -1

but x ≠ -1[as condition is .. x² < 1/6 ] so, x = 1/6

hence, value of x = 1/6