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if ABC is a triangle and tan(A/2), tan(B...

if ABC is a triangle and `tan(A/2), tan(B/2), tan(C/2)` are in H.P. Then find the minimum value of `cot(B/2)`

Text Solution

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`A+B+C=pi`
or `(A)/(2)+(B)/(2)=(pi)/(2)-(C)/(2)`
or `cot((A)/(2)+(B)/(2))=cot((pi)/(2)-(C)/(2))`
or `(cot((A)/(2))cot((B)/(2))-1)/(cot((A)/(2))+cot((B)/(2))=tan""(C)/(2)=(1)/(cot((C)/(2))`
or `cot""(A)/(2)cot""(B)/(2)cot""(C)/(2)=cot""(A)/(2)+cot""(B)/(2)+cot""(C)/(2)`
But `tan""(A)/(2),tan""(B)/(2),tan""(C)/(2)` are in HP
`therefore cot""(A)/(2),cot""(B)/(2),cot""(c)/(2)` are in AP
So, `cot""(A)/(2)+cot""(C)/(2)=2cot""(B)/(2)`
Hence, Eq. (i) becomes
`cot""(A)/(2)cot""(B)/(2)cot""(C)/(2)=3cot""(B)/(2)`
`rArr cot""(A)/(2)cot""(C)/(2)=3`.
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