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

Hint: The inverse trigonometric functions are used to find the missing angles. Here we will use the inverse cosine and cosine function to solve the given expression. Apply the basic concepts of Trigonometry and Inverse Trigonometric Functions.

Complete step-by-step answer:
Take the given expression –
 $ {\cot ^{ - 1}}(1) $
Referring the trigonometric table for the values for cot function, it implies that $ \cot 45^\circ = 1 $
Place the cosine angle in the given expression –
 $ {\cot ^{ - 1}}\left( 1 \right) = {\cot ^{ - 1}}(\cot 45^\circ ) $
Cot inverse and cot function cancel each other.
 $ {\cot ^{ - 1}}\left( 1 \right) = (45^\circ ) $
The above expression can be re-written as –
 $ {\cot ^{ - 1}}\left( 1 \right) = \dfrac{\pi }{4} $
This is the required solution.
So, the correct answer is “ $ \dfrac{\pi }{4} $ ”.

Note: Remember the trigonometric formulas and the correlation between the trigonometric functions to find the unknowns. Remember the trigonometric table for the reference values for different angles for sine, cosine and tangent functions for direct substitution. Also, remember that tangent and cot are inverse of each other.