Let $A,\ B$ and $C$ are the angles of a teiangle and$ an\left( \frac{A}{2} ight) = \frac{1}{3}, an\left( \frac{B}{2} ight) = \frac{2}{3}$. Then, $ an\left( \frac{C}{2} ight)$ is equal to

Question

Let $A,\ B$ and $C$ are the angles of a teiangle and$	an\left( \frac{A}{2} ight) = \frac{1}{3},	an\left( \frac{B}{2} ight) = \frac{2}{3}$. Then, $	an\left( \frac{C}{2} ight)$ is equal to

Dr. Abubacker · Accepted Answer

(d)
Given, $A + B + C = \pi$
$\Rightarrow \ \ \ \frac{A + B}{2} = \frac{\pi}{2} - \frac{C}{2}$
$\Rightarrow \ 	an\left( \frac{A + B}{2} ight) = 	an\left( \frac{\pi}{2} - \frac{C}{2} ight) = \cot\frac{C}{2}$
$\Rightarrow \ \frac{	an{\frac{A}{2} + 	an\frac{B}{2}}}{1 - 	an{\frac{A}{2}	an\frac{B}{2}}} = \cot\frac{C}{2}$
$\Rightarrow \ \ \frac{\frac{1}{3} + \frac{2}{3}}{1 - \frac{1}{3} 	imes \frac{2}{3}} = \cot\frac{C}{2}$
$\left\lbrack \because\ 	an{\frac{A}{2} = \frac{1}{3},	an{\frac{B}{2} = \frac{2}{3}\ (given)}} ightbrack$
$\Rightarrow \ \cot{\frac{C}{2} = \frac{9}{7}}$
$\Rightarrow \ 	an{\frac{C}{2} = \frac{7}{9}}$

Q. Let $A,\ B$ and $C$ are the angles of a teiangle and$\tan\left( \frac{A}{2} \right) = \frac{1}{3},\tan\left( \frac{B}{2} \right) = \frac{2}{3}$. Then, $\tan\left( \frac{C}{2} \right)$ is equal to

Explanation

Reviewed by Dr. Abubacker

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