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

In any triangle `A B C` , prove that following: `\ (cos^2B-cos^2C)/(b+c)+(cos^2C-cos^2A)/(c+a)+(cos^2A-cos^2B)/(a+b)=0`

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Let `frac{a}{sin A}=frac{b}{sin B}=frac{c}{sin C}=k`
Then,
Consider the LHS of the equation `frac{cos ^{2} B-cos ^{2} C}{b+c}+frac{cos ^{2} C-cos ^{2} A}{c+a}+frac{cos ^{2} A-cos ^{2} B}{a+b}=0`.
`LHS=frac{cos ^{2} B-cos ^{2} C}{b+c}+frac{cos ^{2} C-cos ^{2} A}{c+a}+frac{cos ^{2} A-cos ^{2} B}{a+b}`
Now,
`frac{cos ^{2} B-cos ^{2} C}{b+c}=frac{cos ^{2} B-cos ^{2} C}{k(sin B+sin C)}`
`=frac{(cos B+cos C)(cos B-cos C)}{k(sin B+sin C)}`
`(therefore cos ^{2} B-cos ^{2} C.=(cos B+cos C)(cos B-cos C))`
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