Question

Solve for x, 

`2 tan^(−1) x + sin^(−1)((2x)/(1 + x^2)) = 4sqrt3`

Answer 1

Given: `2 tan^(−1) x + sin^(−1)((2x)/(1 + x^2)) = 4sqrt3`

Put x = tan θ ⇒ θ = tan⁻¹ x

`2 tan^(-1) (tan θ) + sin^(-1)((2tan θ)/(1 + tan^2 θ)) = 4sqrt3`

⇒ `2θ + sin^(-1)(sin 2θ) = 4sqrt3`

⇒ `2θ + 2θ = 4sqrt3`

⇒ `4θ = 4sqrt3`

⇒ `θ = (4sqrt3)/4`

`∴ θ = sqrt3`

Now, `x = tan^(-1)x  sqrt3   ...(∵ θ = tan^(-1)x)`

There is no solution for x.