How do you find the exact value of \[{\tan ^{ - 1}}\left( { - 1} \right)\] \[?\]

Hint: To find the exact value of \[{\tan ^{ - 1}}\left( { - 1} \right)\] we have to know some inverse trigonometric properties. The \[\tan \] of a negative value is minus the \[\tan \] of its positive value. So I neglected the negative sign: find \[{\tan ^{ - 1}}\left( 1 \right)\] . Then find \[\tan \] at those values. Since \[\tan x\] is a periodic function with period \[\pi \] . Using the definition of a periodic function, find the exact value of \[{\tan ^{ - 1}}\left( { - 1} \right)\] .

Complete step by step solution:
Given \[{\tan ^{ - 1}}\left( { - 1} \right)\] -----(1)
We know that \[\tan x = \dfrac{{\sin x}}{{\cos x}}\] ------(2)
Since from the equation (1) neglect the negative sign, we get \[{\tan ^{ - 1}}\left( 1 \right) = \dfrac{\pi }{4} \Rightarrow \tan \left( {\dfrac{\pi }{4}} \right) = 1\] .So from the equation (2) we have to find the value of the \[\sin x\] , \[\cos x\] at \[x = \dfrac{\pi }{4}\] .
Hence, we know that \[\sin \left( {\dfrac{\pi }{4}} \right) = \dfrac{1}{{\sqrt 2 }}\] and \[\cos \left( {\dfrac{\pi }{4}} \right) = \dfrac{1}{{\sqrt 2 }}\] ------(3)
Consider \[\tan \left( { - \dfrac{\pi }{4}} \right) = - \tan \left( {\dfrac{\pi }{4}} \right) = - 1\]
  \[ \Rightarrow {\tan ^{ - 1}}\left( { - 1} \right) = - \dfrac{\pi }{4}\] -------(3)
Since \[\tan x\] is a periodic function with period \[\pi \] . By definition of a periodic function, there exist any integer \[n\] , such that
 \[\tan \left( {n\pi - \dfrac{\pi }{4}} \right) = - 1\] for any integer \[n\] .
 \[ \Rightarrow {\tan ^{ - 1}}\left( { - 1} \right) = n\pi - \dfrac{\pi }{4}\]
Since the range of \[{\tan ^{ - 1}}\left( x \right)\] lie in the range \[\left( { - \dfrac{\pi }{2},\dfrac{\pi }{2}} \right)\]
Hence the exact value of \[{\tan ^{ - 1}}\left( { - 1} \right)\] is \[ - \dfrac{\pi }{4}\] .
So, the correct answer is “ \[ - \dfrac{\pi }{4}\] ”.

Note: The inverse of the trigonometric function must be used to determine the measure of the angle. The inverse of the tangent function is read tangent inverse and is also called the arctangent relation. The inverse of the cosine function is read cosine inverse and is also called the arccosine relation. The inverse of the sine function is read sine inverse and is also called the arcsine relation.
The principal value denoted \[{\tan ^{ - 1}}\] is chosen to lie in the range \[\left( { - \dfrac{\pi }{2},\dfrac{\pi }{2}} \right)\] . Hence the exact value of \[{\tan ^{ - 1}}\left( { - x} \right)\] for any value of \[x\] lies in the \[\left( { - \dfrac{\pi }{2},\dfrac{\pi }{2}} \right)\] .Also note that \[\sin ( - x) = - \sin (x)\] , \[\cos ( - x) = \cos (x)\] and \[\tan ( - x) = - \tan (x)\] .