Question

Find the value of ${\tan }^{-1}\left(1\right)$ and ${\tan }^{-1}\left(\tan 1\right)$.

Answer 1

Find the value of the given inverse trigonometric functions

Given: ${\tan }^{-1}\left(1\right)$

We know that the range of $${\tan }^{-1}$$ is $$\left(-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}}\right)$$ and $${\tan }^{-1}\left(\tan x\right)=x$$ if $$x\in \left(-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}}\right)$$.

So,

$$\begin{gathered} {\tan }^{-1}\left(1\right)={\tan }^{-1}\left(\tan \frac{\mathrm{\pi }}{4}\right)[\because \tan \frac{\mathrm{\pi }}{4}=1] \\ \Rightarrow {\tan }^{-1}\left(1\right)=\frac{\mathrm{\pi }}{4} \end{gathered}$$

Now,

Given: ${\tan }^{-1}\left(\tan 1\right)$

We know that the range of $${\tan }^{-1}$$ is $$\left(-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}}\right)$$ and $${\tan }^{-1}\left(\tan x\right)=x$$ if $$x\in \left(-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}}\right)$$.

So,

${\tan }^{-1}\left(\tan 1\right)=1$

Hence, the value of ${\tan }^{-1}\left(1\right)$ is $\frac{\mathrm{\pi }}{4}$ and the value of ${\tan }^{-1}\left(\tan 1\right)$ is $1$.