$\operatorname{cosa}=-\frac{5}{13}\Rightarrow {{\sin }^{2}}a=1-{{\cos }^{2}}a=1-\frac{25}{169}=\frac{144}{169}\Rightarrow \operatorname{sina}=\pm \frac{12}{13}\xrightarrow {a}\operatorname{sina}=\frac{12}{13}$  $\tan \frac{a}{2}=\frac{1-\operatorname{cosa}}{\operatorname{sina}}=\frac{1-\left( \frac{-5}{13} \right)}{\frac{12}{13}}=\frac{\frac{18}{13}}{\frac{12}{13}}=\frac{18}{12}=\frac{3}{2}=1/5$