Question

Solve the following equation for x:

tan⁻¹(x + 1) + tan⁻¹(x − 1) = tan⁻¹`8/31`

Answer 1

Given: tan⁻¹(x + 1) + tan⁻¹(x − 1) = tan⁻¹`8/31`

Take LHS

tan⁻¹(x + 1) + tan⁻¹(x − 1) = tan⁻¹`8/31`

We know that, Formula

tan⁻¹ x + tan-1 y = tan-1 `(x + y)/(1 - xy)`

Thus,

`=> tan^-1  ((x + 1)+(x - 1))/(1 -(x + 1)xx(x - 1)) = tan^-1  8/31`

`=> tan^-1  (2x)/(1-(x^2 - 1)) = tan^-1  8/31`

`=> tan^-1  (2x)/(1 - x^2 + 1) = tan^-1  8/31`

`=> (2x)/(1 - x^2 + 1) = 8/31`

⇒ 62x = 8 − 8x² + 8

⇒ 4x² + 62x − 16 = 0

⇒ 6x² + 31x − 8 = 0

⇒ 4x(x + 8) − 1(x + 8) = 0

⇒ (4x − 1)(x + 8) = 0

⇒ 6x + 1 = 0 or x − 1 = 0

⇒ x = `1/4` or x = −8

Since,

x = `1/4` ∈ `(-sqrt2, sqrt2)`

So, 

x = `1/4` is the root of the given equation

Therefore, 

x = `1/4`