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Question
Mark the correct alternative in each of the following:
In any ∆ABC, 2(bc cosA + ca cosB + ab cosC) =
Options
\[abc\]
\[a + b + c\]
\[a^2 + b^2 + c^2\]
\[\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}\]
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Solution
Using cosine rule, we have
\[2\left( bc\cos A + ca\cos B + ab\cos C \right)\]
\[ = 2bc\left( \frac{b^2 + c^2 - a^2}{2bc} \right) + 2ca\left( \frac{c^2 + a^2 - b^2}{2ca} \right) + 2ab\left( \frac{a^2 + b^2 - c^2}{2ab} \right)\]
\[ = b^2 + c^2 - a^2 + c^2 + a^2 - b^2 + a^2 + b^2 - c^2 \]
\[ = a^2 + b^2 + c^2\]
Hence, the correct answer is option (c).
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