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.