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In any triangle A B C , prove that: (1+c...

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

Text Solution

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`(1 + cos (A - B) cos C)/(1 + cos (A - C) cos B) = (1 - cos (A - B) cos (A + B))/(1 - cos (A - C) cos (A + C))`
`= (1- (cos^(2) A - sin^(2) B))/(1 - (cos^(2) A - sin^(2) C))`
`= (sin^(2) A + sin^(2)B)/(sin^(2) A + sin^(2) C)`
`= (a^(2) + b^(2))/(a^(2) + c^(2))` [using a = 2 R sin A, etc]
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