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In DeltaABC, prove that: cot\ A/2+cot\ ...

In `DeltaABC`, prove that: `cot\ A/2+cot\ B/2+cot\ C/2=((a+b+c)^(2))/(4Delta)`

Text Solution

Verified by Experts

LHS `=cotA/2+cotB/2+cotC/2`
`=sqrt((s(s-a))/((s-b)(s-c))+sqrt((s(s-b))/((s-a)(s-c)) + sqrt((s(s-c))/((s-a)(s-b))`
`=(sqrt(s)[(s-a)+(s-b)+(s-c)])/(sqrt((s-a)(s-b)(s-c)))`
`(sqrt(s)[(s-a)+(s-b)+(s-c)])/(sqrt((s-a)(s-b)(s-c)))`
`(sqrt(s).sqrt(s)[3s-(a+b+c)])/(sqrt(s)(s-a)(s-b)(s-c))`
`(s.(3s-2s))/(Delta)=s^(2)/Delta=(4s^(2))/(4Delta)`
`=(2s)^(2)/(4Delta)=(a+b+c)^(2)/(4Delta)`= RHS Hence Proved.
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