Question

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

Answer 1

Hint : You should know that the range of ${\tan ^{ - 1}}$ is $\left( { - \dfrac{\pi }{2},\dfrac{\pi }{2}} \right)$ and ${\tan ^{ - 1}}\left( {\tan x} \right) = x$ if $x \in \left( { - \dfrac{\pi }{2},\dfrac{\pi }{2}} \right)$ . To find the value of ${\tan ^{ - 1}}\left( 1 \right)$ we need to check that at which value of $x$ the $\tan x$ is $1$ . After this we can easily find the value of ${\tan ^{ - 1}}\left( 1 \right)$ without substituting any value to the function. And in the case of ${\tan ^{ - 1}}\left( {\tan 1} \right)$ , ${\tan ^{ - 1}}$ simply cancel out by $\tan$ .

Complete step-by-step answer :
In these type of questions we have to keep one thing in mind that we have to cancel out ${\tan ^{ - 1}}$ by $\tan$
. To find the value of ${\tan ^{ - 1}}\left( 1 \right)$ , let $y = {\tan ^{ - 1}}\left( 1 \right)$
We have to make that equation and suitable conditions by which we can do this, so if $1$ is given then it is $\tan \dfrac{\pi }{4}$ as the value of $\tan \dfrac{\pi }{4}$ is $1$ . Therefore we can write it as
$y = {\tan ^{ - 1}}\left( {\tan \dfrac{\pi }{4}} \right)$
By taking inverse of tan to the other side we get
$\tan y = \left( {\tan \dfrac{\pi }{4}} \right)$
What happens now, the tan terms will cancel out so simply we will get $\dfrac{\pi }{4}$ as a simple answer that is we are only left with $y = \dfrac{\pi }{4}$ and $y = {\tan ^{ - 1}}\left( 1 \right)$ which means that the value of ${\tan ^{ - 1}}\left( 1 \right)$ is $\dfrac{\pi }{4}$
Again, to find the value of the ${\tan ^{ - 1}}\left( {\tan 1} \right)$ , let $y = {\tan ^{ - 1}}\left( {\tan 1} \right)$
By taking inverse of tan to the other side we get
$\tan y = \tan 1$
In this case ${\tan ^{ - 1}}$ simply will cancel out by $\tan$ and we got one as an answer. Or we can say that the tan on both the sides will cancel out and we are left with $y = 1$ and $y = {\tan ^{ - 1}}\left( {\tan 1} \right)$ which means that the value of ${\tan ^{ - 1}}\left( {\tan 1} \right) = 1$ .
Hence, the value of ${\tan ^{ - 1}}\left( 1 \right)$ is $\dfrac{\pi }{4}$ and the value of ${\tan ^{ - 1}}\left( {\tan 1} \right) = 1$

Note : Keep in mind that ${\tan ^{ - 1}}\left( {\tan x} \right) = x$ if $x \in \left( { - \dfrac{\pi }{2},\dfrac{\pi }{2}} \right)$ . The inverse tangent of $1$ is $\dfrac{\pi }{4}$ . tan is an increasing function for all $x$ between $0$ and $\dfrac{\pi }{2}$ . The value of the inverse trigonometric function which lies in the range of the principal branch is its principal value.