From the sum identity of the tangent function given by
[tex]\tan (\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\cdot\tan\beta}[/tex]
and by comparing this fomula with our expression, we can note that
[tex]\begin{gathered} \alpha=\frac{5\pi}{14} \\ \text{and} \\ \beta=\frac{\pi}{14} \end{gathered}[/tex]
Then by substituting these values into the formula, we get
[tex]\tan (\frac{5\pi}{14}+\frac{\pi}{14})=\frac{\tan\frac{5\pi}{14}+\tan\frac{\pi}{14}}{1-\tan\frac{5\pi}{14}\cdot\tan\frac{\pi}{14}}[/tex]
Since
[tex]\frac{5\pi}{14}+\frac{\pi}{14}=\frac{6\pi}{14}=\frac{3\pi}{7}[/tex]
the answer is:
[tex]tan\frac{3\pi}{7}[/tex]
