On inverse trigonometric functions

Question

Can we always write tan^-1 (x) as cot^-1(1/x),sin^-1(x) as cosec^-1(x) and sec^-1(x) as cos^-1 (x) or are there any restrictions on this fact because of domain ?

Answer 1

The principle values for the inverse functions were chosen exactly with the intent that $$\operatorname{Arcsin}\left(\frac 1x\right) = \operatorname{Arccsc}(x),\quad \operatorname{Arccsc}\left(\frac 1x\right) = \operatorname{Arcsin}(x)$$ $$\operatorname{Arccos}\left(\frac 1x\right) = \operatorname{Arcsec}(x),\quad \operatorname{Arcsec}\left(\frac 1x\right) = \operatorname{Arccos}(x)$$ everywhere in their domains except for $x = 0$.

However, it is common to take $\left(-\frac \pi 2, \frac \pi 2\right)$ as the range of $\operatorname{Arctan}$, but $(0,\pi)$ as the range of $\operatorname{Arccot}$. Therefore

$$\operatorname{Arctan}\left(\frac 1x\right) = \operatorname{Arccot}(x),\quad \operatorname{Arccot}\left(\frac 1x\right) = \operatorname{Arctan}(x)$$ holds for $x > 0$, but $$\operatorname{Arctan}\left(\frac 1x\right) = \operatorname{Arccot}(x) - \pi,\quad \operatorname{Arccot}\left(\frac 1x\right) = \operatorname{Arctan}(x) + \pi$$ holds for $x < 0$.