Answer
The exact value of the provided expression is $-\frac{1}{2}$.
Work Step by Step
By putting the values of the trigonometric functions, we evaluate as follows:
$\begin{align}
& {{\cos }^{2}}\frac{\pi }{4}-{{\tan }^{2}}\frac{\pi }{4}={{\left( \frac{1}{\sqrt{2}} \right)}^{2}}-{{\left( 1 \right)}^{2}} \\
& =\frac{1}{2}-1 \\
& =\frac{1-2}{2} \\
& =-\frac{1}{2}
\end{align}$
Hence, the exact value of the provided expression is $-\frac{1}{2}$.
