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If A+B+C=pi, prove that : cot, A/2+ cot,...

If `A+B+C=pi`, prove that : `cot, A/2+ cot, B/2 + cot, C/2 = cot, A/2 cot, B/2 cot, C/2`

Text Solution

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(a) `A+B+C=pi` or `3A+3B+3C=3pi`
`rARr P+Q+R=3pi`,
Where `P=3A,Q=3B,=3C`
`rArr tan P+tan Q+tanR=tan P tan Q tan R`
`rArr tan 3A+tan 3BJ+tan3C=tan3A tan 3B tan 3C`
(b) In triangle `ABC,(A)/(2)+(B)/(2)+(C)/(2=(pi)/(2)`
or `(A)/(2)+(B)/(2)=(pi)/(2)-(C)/(2)`
or `tan((A)/(2)+(B)/(2))=tan((pi)/(2)-(C)/(2))`
or `(tan""(A)/(2)+tan""(B)/(2))/(1-tan""(A)/(2)tan""(B)/(2))=cot(C)/(2)`
or `(cot((B)/(2))+cot((A)/(2))/(cot((A)/(2))cot((B)/(2))-1)=cot(C)/(2)`
or `cot""(A)/(2)+cot""(B)/(2)+cot""(C)/(2)=cot""(A)/(2) cot""(B)/(2)cot""(C)/(2)`
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