2.3 ${\tan ^{ - 1}}x$:

2.1 ${\sin ^{ - 1}}x$:
2.2 ${\cos ^{ - 1}}x$:
2.3 ${\tan ^{ - 1}}x$:
2.4 ${\text{cose}}{{\text{c}}^{ - 1}}x$:
2.5 ${\sec^{ - 1}}x$
2.6 ${\cot^{ - 1}}x$
2.7 Summary

3.1 Property 1
3.2 Property 2
3.3 Property 3
3.4 Property 4
3.5 Property 5
3.6 Property 6
3.7 Property 7
3.8 Property 8
3.9 Property 9
3.10 Property 10
3.11 Property 11
3.12 Property 12
3.13 Property 13

We denote the inverse of tan function by ${\tan ^{ - 1}}x$ (arc $\tan x$) function.

From our previous discussion, the inverse of a function is possible only when it is one-one and onto i.e., bijective. So, we have to choose a certain set of domain and range to consider ${\tan ^{ - 1}}x$ as a function.

As we know, the domain of tan function is $R - \{ (2n + 1)\frac{\pi }{2}\} $ and range is $R$.

If we restrict its domain to $\left( {-\frac{\pi }{2}, \frac{\pi }{2}} \right),\left( { - \frac{{2\pi }}{2}, - \frac{\pi }{2}} \right),\left( {\frac{\pi }{2},\frac{{3\pi }}{2}} \right)$... etc and its range is $R$ then it becomes one-one and onto.

We conclude that the inverse of tan function exists with in all the above mentioned set of domain and range of $R$ out of which domain $\left( {-\frac{\pi }{2}, \frac{\pi }{2}} \right)$ is the principle value set.

We can write the inverse of tan function as, $$y={\tan ^{ - 1}}x$$ where domain becomes $R$ and range becomes $\left( {-\frac{\pi }{2}, \frac{\pi }{2}} \right)$.