$(b + c) cos A + (c + a) cos B + (a + b) cos C = a + b + c$ .

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Solution

Taking LHS
$= (b + c) c o s A + (c + a) c o s B + (a + b) c o s C = b cos A + c cos A + c cos B + a cos B + a cos C + b cos C = (b cos A + a cos B) + (c cos A + a cos C) + (b cos C + c cos B) = a + b + c$
Hence proved

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