Answer
The exact value of the expression is $\frac{\sqrt{2}+1}{2}$.
Work Step by Step
Let us consider the following equation.
$x=\sin \frac{\pi }{4}\cos 0-\sin \frac{\pi }{6}\cos \pi $.
Put $\frac{\sqrt{2}}{2}$ for $\sin \frac{\pi }{4}$ , $1$ for $\cos 0$ , $\frac{1}{2}$ for $\sin \frac{\pi }{6}$ and $-1$ for $\cos \pi $.
$\begin{align}
& x=\frac{\sqrt{2}}{2}\left( 1 \right)-\left( \frac{1}{2} \right)\left( -1 \right) \\
& =\frac{\sqrt{2}}{2}+\frac{1}{2} \\
& =\frac{\sqrt{2}+1}{2}
\end{align}$
