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In any triangleABC prove that(a+b+c)(tanA/2+tanB/2)=2cotC/2

In any triangleABC prove that(a+b+c)(tanA/2+tanB/2)=2cotC/2

Grade:11

2 Answers

Arun
25763 Points
2 years ago
Dear Satya
 
$$2s=a+b+c$$ We know, $$\tan (A/2) = \sqrt{\frac{(s-b)(s-c)}{s(s-a)}}$$ Similarly, $$\tan (B/2) = \sqrt{\frac{(s-a)(s-c)}{s(s-b)}}$$ Now, $$(a+b+c)(\tan (A/2)+\tan (B/2)=(a+b+c)(\sqrt{\frac{(s-b)(s-c)}{s(s-a)}}+\sqrt{\frac{(s-a)(s-c)}{s(s-b)}})$$ $$=2s\sqrt{\frac{s-c}{s}}(\sqrt{\frac{s-b}{s-a}}+\sqrt{\frac{s-a}{s-b}})$$ $$=2s\sqrt{\frac{s-c}{s}}(\frac{s-b+s-a}{\sqrt{(s-b)(s-a}})$$ $$=2c\sqrt{\frac{s(s-c)}{(s-a)(s-b)}}$$ $$=2c\tan (C/2)$$
 
Hope it helps
In case of any difficulty please feel free to ask
 
Regards
Arun (askIITians forum expert)
Deepak Kumar Shringi
askIITians Faculty 4405 Points
2 years ago
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