Mathematics - Trigonometric Ratios & Identities Question with Solution | TestHub

$\begin{array}{rl}& \mathrm{A}+\mathrm{B}+\mathrm{C}=180^{\circ }\text{ (Angle sum property of a triangle) }\\ & \Rightarrow \mathrm{A}+\mathrm{B}=180^{\circ }-\mathrm{C}\\ & \Rightarrow \frac{A+B}{2}=90^{\circ }-\frac{C}{2}\\ & \Rightarrow \tan \left(\frac{A}{2}+\frac{B}{2}\right)=\tan \left(90^{\circ }-\frac{C}{2}\right)\\ & \Rightarrow \frac{\tan A/2+\tan B/2}{1-\tan A/2\tan B/2}=\cot C/2\\ & \Rightarrow \frac{\frac{1}{3}+\frac{2}{3}}{1-\frac{1}{3}\times \frac{2}{3}}=\cot C/2\Rightarrow \frac{1}{1-\frac{2}{9}}=\cot C/2\\ & \Rightarrow \frac{9}{7}=\cot C/2\Rightarrow \tan C/2=7/9\end{array}$