Question

Write the following function in the simplest form:

`tan^(-1)  (sqrt(1 + x^2) - 1)/x, x ≠ 0`

Answer 1

`tan^(-1)  (sqrt(1 + x^2) - 1)/x`

Put x = tan θ ⇒ θ = tan^–1 x

∴ `tan^(-1)  (sqrt(1 + x^2) - 1)/x`

= `tan^(-1) ((sqrt(1 + tan^2 θ) - 1)/tan θ)`

= `tan^(-1) ((sec θ - 1)/tan θ)`

= `tan^(-1) ((1 - cos θ)/sin θ)`

= `tan^(-1) ((2 sin^2  θ/2)/(2 sin  θ/2  cos  θ/2))`

= `tan^(-1) (tan  θ/2)`

= `θ/2`

= `1/2 tan^(-1) x`