Question

lf in $△ABC,6\cos C+7\cos B=5$ and $7\cos A+5\cos C=6$, then the value of $\tan \frac{B}{2}\tan \frac{C}{2}$ is

• A. $\frac{1}{3}$
• B. $\frac{2}{9}$
• C. $\frac{4}{9}$
• D. $\frac{5}{9}$

Answer 1

The correct option is C $\frac{4}{9}$

By applying projection rule we get,
$a=b\cos C+c\cos B\to \left(i\right)$
$b=c\cos A+a\cos C\to \left(ii\right)$
Hence, $a=5,b=6,c=7$ and $s=9$
Also,
$\tan \frac{A}{2}=\sqrt{\frac{(s-b)(s-c)}{s(s-a)}}$
$\tan \frac{B}{2}=\sqrt{\frac{(s-c)(s-a)}{s(s-b)}}$
$\tan \frac{C}{2}=\sqrt{\frac{(s-a)(s-b)}{s(s-c)}}$
Now, $\tan \frac{B}{2}\tan \frac{C}{2}=\frac{s-a}{s}=\frac{9-5}{9}=\frac{4}{9}$