$2 (b c c o s A + c a c o s B + a b c o s C) = a^{2} + b^{2} + c^{2}$

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Solution

$2 (b c c o s A + c a c o s B + a b c o s C) = a^{2} + b^{2} + c^{2} L H S = 2 b c c o s A + 2 c a c o s B + 2 a b c o s C = 2 b c \frac{b^{2} + c^{2} - a^{2}}{2 b c} + 2 c a \frac{a^{2} + c^{2} - b^{2}}{2 c a} + 2 a b \frac{a^{2} + b^{2} - c^{2}}{2 a b} = b^{2} + c^{2} - a^{2} + a^{2} + c^{2} - b^{2} + a^{2} + b^{2} - c^{2} = a^{2} + b^{2} + c^{2} = R H S$

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