Question

Solve the following equation for x:

tan⁻¹(x −1) + tan⁻¹x tan⁻¹(x + 1) = tan⁻¹3x

Answer 1

We know

`tan^-1x+tan^-1y=tan^-1((x+y)/(1-zy))and tan^-1x-tan^-1y=tan^-1((x-y)/(1+xy))`

∴ tan⁻¹(x −1) + tan⁻¹x tan⁻¹(x + 1) = tan⁻¹3x

⇒ `tan^-1{(x+1+x-1)/(1-(x+1)xx(x+1))}=tan^-1 3x-tan^-1x`

⇒ `tan^-1((2x)/(2-x^2))(=tan^-1((3x-x)/(1+3x^2))`

⇒ `(2x)/(2-x^2)=(2x)/(1+3x^2)`

⇒ `2-x^2=1+3x^2`

⇒ 4x² - 1 = 0

⇒ `x^2=1/4`

⇒ `x=+-1/2`