Question

In $\Delta$ $ABC$ the sides opposite to angles $A,B,C$ are denoted by $a,b,c$ respectively.

If $\frac{r}{r_1}=\frac{1}{2}$

then the value of $tan\frac{A}{2}(tan\frac{B}{2}+tan\frac{C}{2})$is equal to :

• A. $2$
• B. $\frac{1}{2}$
• C. $1$
• D. None of these

Answer 1

Answer: (B)

$\frac{1}{2}$

Given, $\frac{r}{r_1}=\frac{1}{2}\Rightarrow \frac{\frac{\Delta /s}{\Delta }}{s-a}=\frac{1}{2}$
$\Rightarrow \frac{s-a}{s}=\frac{1}{2}$
$\Rightarrow s=2a$

Now, $\tan \frac{A}{2}(\tan \frac{B}{2}+\tan \frac{c}{2})$
$=\frac{\Delta }{s(s-a)}(\frac{\Delta }{s(s-b)}+\frac{\Delta }{s(s-c)})$
$\Rightarrow \frac{\Delta }{s(s-a)}\frac{\Delta }{s}\frac{a}{(s-b)(s-c)}=\frac{a}{s}=\frac{1}{2}$