JEE Main & Advanced Sample Paper JEE Main - Mock Test - 5

Let A, B, and C are the angles of a plain triangle and \[\tan \frac{A}{2}=\frac{1}{3},\]\[\tan \frac{B}{2}=\frac{2}{3}.\] Then \[\tan \frac{C}{2}\] is equal to

A) \[7/9\]

B) \[2/9\]

C) \[1/3\]

D) \[2/3\]

Correct Answer: A

Solution :

\[A+B+C=\pi \]

\[\therefore \,\,\,\,\tan \left( \frac{A+B}{2} \right)=\tan \left( \frac{\pi }{2}-\frac{C}{2} \right)\]

\[\Rightarrow \,\,\,\,\frac{\tan \frac{A}{2}+\tan \frac{B}{2}}{1-\tan \frac{A}{2}.\tan \frac{B}{2}}=\cot \,\frac{C}{2}\]

\[\Rightarrow \,\,\frac{\frac{1}{3}+\frac{2}{3}}{1-\frac{1}{3}.\frac{2}{3}}=\frac{9}{7}=\cot \,\frac{C}{2}\]

\[\therefore \,\tan \,\frac{C}{2}=\frac{7}{9}\].