$Solve:{\text{ tan}}\left( {\pi - {{\tan }^{ - 1}}z} \right) = z$

Hint: This is an inverse-trigonometry question so one must know the identities related to it. In this question the key observations are, as the angle $\left( {\pi - \theta } \right)$ lies in the second quadrant if $\theta $ is assumed to be an acute angle. And also ${\text{tan}}\left( {{{\tan }^{ - 1}}z} \right) = z$ is used.

Complete step-by-step answer:
The given equation is,
${\text{tan}}\left( {\pi - {{\tan }^{ - 1}}z} \right) = z$
Let, $\theta = {\tan ^{ - 1}}z$
$ \Rightarrow \tan \theta = z$
$\therefore $ The equation becomes
${\text{tan}}\left( {\pi - \theta } \right) = z$
When $\theta $ is an acute angle, then the equation becomes
$ - \tan \theta = z$
Putting the value of $\theta $,
$ \Rightarrow - \tan \left( {{{\tan }^{ - 1}}z} \right) = z$
$\because {\text{ }}\tan ({\tan ^{ - 1}}x) = x$
$\therefore $ The equation can be further written as,
$ - z = z$
Adding z to both sides,
$2z = 0$
Dividing 2 on both sides,
$ \Rightarrow z = 0$
$\therefore $ The final answer is $z = 0$ the solution of the given equation.

Note: This is trigonometry-based question and in order to solve this the identities related to it must be known. The equation should be solved carefully. The quadrant in which it lies must be seen carefully and according to that the sign of the trigonometric-function is determined. Remember that in first quadrant all trigonometry functions are positive, in second quadrant only $\sin \theta $ is positive, in third quadrant only $\tan \theta $ is positive and in fourth quadrant only $\cos \theta $ is positive. Calculations should be done carefully to avoid any mistake. After the final answer is found out it can be checked that whether it satisfy the original equation given in the question by simply substituting its value in the equation and if does not satisfy the equation then the solution must be rechecked. The equation should be solved in accordance with the identities which would result in correct solution. Always try to solve the question step by step so that the wrong step can be determined and changed.