Let $f\left( \theta  \right) = \begin{array}{*{20}{c}}
1&{\cos \theta }&1\\
{ – \sin \theta }&1&{ – \cos \theta }\\
{ – 1}&{\sin \theta }&1
\end{array}$

$ = \left( {1 + \sin \theta \cos \theta } \right) – \cos \theta \left( {\sin \theta  – \cos \theta } \right) + 1\left( { – {{\sin }^2}\theta  + 1} \right)$

$ = 1 + \sin \theta \cos \theta  + \sin \theta \cos \theta  + {\cos ^2}\theta  – {\sin ^2}\theta  + 1$

$ = 2 + 2\sin \theta \cos \theta  + \cos 2\theta $

$ = 2 + \sin 2\theta  + \cos 2\theta \,\,\,\,\,\,\,\,\,……\left( 1 \right)$

Now, maximum value of $(1)$

is $2 + \sqrt {{1^2} + {1^2}}  = 2 + \sqrt 2 $

and minimum value of $(1)$ is

$2 – \sqrt {{1^2} + {1^2}}  = 2 – \sqrt 2 $.