Mark the correct alternative in each of the following:
Question:

Mark the correct alternative in each of the following:

In any ∆ABC, 2(bc cosA + ca cosB + ab cosC) =

(a) $a b c$

(b) $a+b+c$

(c) $a^{2}+b^{2}+c^{2}$

(d) $\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}$

Solution:

Using cosine rule, we have

$2(b c \cos A+c a \cos B+a b \cos C)$

$=2 b c\left(\frac{b^{2}+c^{2}-a^{2}}{2 b c}\right)+2 c a\left(\frac{c^{2}+a^{2}-b^{2}}{2 c a}\right)+2 a b\left(\frac{a^{2}+b^{2}-c^{2}}{2 a b}\right)$

$=b^{2}+c^{2}-a^{2}+c^{2}+a^{2}-b^{2}+a^{2}+b^{2}-c^{2}$

$=a^{2}+b^{2}+c^{2}$

Hence, the correct answer is option (c).

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