# Alfaphication of Tan

**Geometry**Level pending

If we are given \(\cot { \alpha } =\frac { 1 }{ 2 } ,\sec { \beta } =\frac { -5 }{ 3 } \) for \(\pi <\alpha <\frac { 3\pi }{ 2 } ,\frac { \pi }{ 2 } <\beta <\pi \). Then let \(\tan { (\alpha +\beta )=\varsigma } \) and \(p\) equals to the number of quadrant in which \(\alpha +\beta \) terminates. If \(\varsigma +p=\frac { m }{ n } \) for coprime integers \(m\) and \(n\).Find \(m+n\).