How do you evaluate ${{\tan }^{-1}}\left( -1 \right)$ without a calculator?

Hint: We have been given an inverse trigonometric function, that is, the inverse of tangent of 1. In order to find its value without using a calculator, we shall first assume is equal to some angle $\theta $. Then we shall transfer the tangent function to the left-hand side and find the principal as well as general solutions of this angle $\theta $.

Complete step by step solution:
In order to solve this problem, we must have prior knowledge of the six trigonometric functions namely, sine, cosine, tangent, cosecant, cotangent and secant functions. We must also be well versed in the graphs of these functions so that we can remember the principal values at some of the important angles.
Let $\theta ={{\tan }^{-1}}1$
We shall transfer inverse of tangent function to the left-hand side and get,
$\Rightarrow \tan \theta =1$
We know that $\tan \dfrac{\pi }{4}=1$.
Now, comparing these, we shall find all the values of angle $\theta $ for which its tangent is equal to 1.
$\Rightarrow \theta =\dfrac{\pi }{4},\dfrac{5\pi }{4},\dfrac{9\pi }{4}......$ and so on
$\Rightarrow \theta =n\pi +\dfrac{\pi }{4}$ , where $n$ is an integer.
The principal values are $\dfrac{\pi }{4}$ and $\dfrac{5\pi }{4}$ only.
Therefore, ${{\tan }^{-1}}\left( -1 \right)=n\pi +\dfrac{\pi }{4}$.

Note:
The principal solutions of any trigonometric function or trigonometric equation are the set of values lying in the interval $\left[ 0,2\pi \right)$. All other values which can be determined by putting various values of integer ‘n’ in a defined set are called the general solutions of that trigonometric equation. However, only the principal values of the trigonometric functions are usually in use in day-to-day mathematics.