Question

If `3sin^-1((2x)/(1 + x^2)) - 4cos^-1((1 - x^2)/(1 + x^2)) + 2tan^-1((2x)/(1 - x^2)) = pi/3`, then x is equal to ______ 

Options

• `sqrt3`

• `1/sqrt3`

• `-1/sqrt3`

• `-sqrt3/4`

Answer 1

If `3sin^-1((2x)/(1 + x^2)) - 4cos^-1((1 - x^2)/(1 + x^2)) + 2tan^-1((2x)/(1 - x^2)) = pi/3`, then x is equal to `underline(1/sqrt3)`.

Explanation:

Given, `3sin^-1((2x)/(1 + x^2)) - 4cos^-1((1 - x^2)/(1 + x^2)) + 2tan^-1((2x)/(1 - x^2)) = pi/3`

Putting x = tan θ, we get

`3sin^-1((2tantheta)/(1 + tan^2theta)) - 4cos^-1((1 - tan^2theta)/(1 + tan^2theta)) + 2tan^-1((2tantheta)/(1 - tan^2theta)) = pi/3`

⇒ `3sin^-1(sin2theta) - 4cos^-1(cos2theta) + 2tan^-1(tan2theta) = pi/3`

⇒ `3(2theta) - 4(2theta) + 2(2theta) = pi/3`

⇒ `6theta - 8theta + 4theta = pi/3`

⇒ `2theta = pi/3`

⇒ `theta = pi/6`

⇒ `tan^-1x = pi/6`

⇒ `x = tan  pi/6`

⇒ `x = 1/sqrt3`