Answer
The exact value of the provided expression is $\frac{7}{2}$.
Work Step by Step
By putting the values of the trigonometric functions, we evaluate as follows:
$\begin{align}
& \sin \frac{\pi }{6}+{{\tan }^{2}}\frac{\pi }{3}=\frac{1}{2}+{{\left( \sqrt{3} \right)}^{2}} \\
& =\frac{1}{2}+3 \\
& =\frac{1+6}{2} \\
& =\frac{7}{2}
\end{align}$
Hence, the exact value of the provided expression is $\frac{7}{2}$.
