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  4. In triangle A B C if tan (A/2) tan (B/2)+ tan (B/2) tan (C/2)=(2/3), then a+c=

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Q. In $\triangle A B C$ if $\tan \frac{A}{2} \tan \frac{B}{2}+\tan \frac{B}{2} \tan \frac{C}{2}=\frac{2}{3}$, then $a+c=$

Trigonometric Functions

Solution:

$\tan \frac{A}{2} \tan \frac{B}{2}+\tan \frac{B}{2} \tan \frac{C}{2}=\frac{2}{3}$
$\Rightarrow \sqrt{\frac{(s-b)(s-c)}{s(s-a)}} \sqrt{\frac{(s-a)(s-c)}{s(s-b)}}$
$+\sqrt{\frac{(s-a)(s-c)}{s(s-b)}} \sqrt{\frac{(s-a)(s-b)}{s(s-c)}}=\frac{2}{3}$
$\Rightarrow \frac{s-c}{s}+\frac{s-a}{s}=\frac{2}{3}$
$\Rightarrow 3(2 s-a-c)=2 s$
$\Rightarrow 3 b=a+b+c$
$\Rightarrow a+c=2 b$