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In any triangle A B C , prove that follo...

In any triangle `A B C` , prove that following : `c/(a+b)=(1-tan(A/2)tan(B/2))/(1+tan(A/2)tan(B/2))`

Text Solution

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`(c)/(a + b) = (2R sin C)/(2R (sin A + sin B)) = (sin (A + B))/(sin A + sin B)`
`= (2 sin.(A + B)/(2) cos.(A + B)/(2))/(2 sin.(1)/(2) (A + B) cos.(1)/(2) (A - B))`
`= (cos.(1)/(2) (A + B))/(cos.(1)/(2) (A - B))`
`= (cos.(1)/(2) A cos.(1)/(2) B - sin.(1)/(2) A sin.(1)/(2) B)/(cos.(1)/(2) A cos.(1)/(2) B + sin.(1)/(2) A sin.(1)/(2)B)`
`= (1 - tan.(1)/(2) A tan.(1)/(2) B)/(1 + tan.(1)/(2) A tan.(1)/(2) B)` [dividing numerator and denominator by `cos.(1)/(2) A cos.(1)/(2) B`]
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