If $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are the angles of a triangle with $ an \frac{A}{2}=\frac{1}{3}, an \frac{B}{2}=\frac{2}{3}$ then the value of $ an \frac{C}{2}$ is

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If $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are the angles of a triangle with $	an \frac{A}{2}=\frac{1}{3}, 	an \frac{B}{2}=\frac{2}{3}$ then the value of $	an \frac{C}{2}$ is

Aicharya · Accepted Answer

$\begin{aligned} & A+B+C=\pi \ 	herefore \quad & 	an \left(\frac{A+B}{2}ight)=	an \left(\frac{\pi}{2}-\frac{C}{2}ight) \ \Rightarrow & \frac{	an \frac{A}{2}+	an \frac{B}{2}}{1-	an \frac{A}{2} 	an \frac{B}{2}}=\cot \frac{C}{2} \ \Rightarrow & \frac{\frac{1}{3}+\frac{2}{3}}{1-\left(\frac{1}{3}ight)\left(\frac{2}{3}ight)}=\cot \frac{C}{2}\end{aligned}$ $\begin{aligned} & \Rightarrow \frac{9}{7}=\cot \frac{C}{2} \ & \Rightarrow 	an \frac{C}{2}=\frac{7}{9}\end{aligned}$

If $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are the angles of a triangle with $\tan \frac{A}{2}=\frac{1}{3},…

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