Question

In a $∆\mathrm{ABC}$ is $5\cos C+6\cos B=4$ and $6\cos A+4\cos C=5$, then $\tan \frac{A}{2}\tan \frac{B}{2}=$

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

Answer 1

The correct option is A

$\frac{1}{5}$

Explanation for the correct option.

Finding the $\tan \frac{A}{2}\tan \frac{B}{2}$:

Given,

$5\cos C+6\cos B=4$ and $6\cos A+4\cos C=5$.

These are in the form of $b\cos C+c\cos A+a,c\cos A+a\cos C=b$.

On comparing these equation we get,

$a=4,b=5,c=6$

$\begin{array}{rcl}s& =& \frac{a+b+c}{2}\\ & =& \frac{15}{2}\end{array}$

$\begin{array}{rcl}\tan \frac{A}{2}\tan \frac{B}{2}& =& \sqrt{\frac{\left(s-b\right)(s-c)}{s(s-a)}}\times \sqrt{\frac{(s-a)(s-c)}{s(s-b)}}\\ & =& \sqrt{\frac{{(s-c)}^2}{s^2}}\\ & =& \frac{(s-c)}{s}\\ & =& \frac{{\displaystyle \frac{15}{2}}-6}{{\displaystyle \frac{15}{2}}}\\ & =& \frac{3}{15}\\ & =& \frac{1}{5}\end{array}$

Hence, the correct option is A.