Question

In $\Delta ABC,$ if $s=\frac{a+b+c}{2},$ then $\left(b{\cos }^2\frac{C}{2}+c{\cos }^2\frac{B}{2}\right)$ is equal to

• A. $s$
• B. $2s$
• C. $4s$
• D. $3s$

Answer 1

Answer: (A)

$b{\cos }^2\frac{C}{2}+c{\cos }^2\frac{B}{2}$
$=b{\left(\sqrt{\frac{s(s-c)}{ab}}\right)}^2+c{\left(\sqrt{\frac{s(s-b)}{ca}}\right)}^2$
$=b\left(\frac{s(s-c)}{ab}\right)+c\left(\frac{s(s-b)}{ca}\right)$
$=\frac{s^2-sc+s^2-sb}{a}=\frac{2s^2-s(b+c)}{a}$
$=\frac{2s^2-s(2s-a)}{a}$
$=\frac{2s^2-2s^2+sa}{a}=s$