Question

Write `tan^(-1) x, x gt 0` in the form of other inverse trigonometric function

Answer 1

For `theta = tan^(-1) x, x gt 0 " or " tan theta = x`, we have following right-angled triangle.
From the figure,

`theta = sin^(-1) (x)/(sqrt(1 + x^(2)))`
`= cos^(-1) (1)/(sqrt(1 + x^(2)))`
`= cot^(-1) (1)/(x)`
`= sec^(-1) sqrt(1 + x^(2))`
`= cosec^(-1) (sqrt(1 + x^(2)))/(x)`