Question

Write the function in the simplest form: `tan^(-1)  1/(sqrt(x^2 - 1)), |x| > 1`

Answer 1

`tan^(-1)  1/(sqrt(x^2 - 1)`, |x| > 1

Put x = cosec θ ⇒ θ = cosec⁻¹ x

`:. tan^(-1)  1/(sqrt(x^2 - 1)) = tan^(-1)  1/(sqrt(cosec^2 theta - 1))`

`= tan^(-1) (1/ cot theta) = tan^(-1) (tan theta)`

`= theta = cosec^(-1) x = pi/2 - sec^(-1) x`

`[cosec^(-1) x + sec^(-1) x = pi/2]`