Mark the correct alternative in each of the following :

$I n a n y Δ A B C, a (b c o s C - c c o s B) =$

$a^{2}$

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$b^{2} - c^{2}$

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$0$

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

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Solution

The correct option is B
$b^{2} - c^{2}$

$\Rightarrow a (b c o s C - c c o s B) = a b c o s C - a c c o s B Putting the values of cos B and cos C, we get = a b [\frac{a^{2} + b^{2} - c^{2}}{2 a b}] - a c [\frac{c^{2} + a^{2} - b^{2}}{2 a c}] = \frac{a^{2} + b^{2} - c^{2}}{2} - \frac{c^{2} + a^{2} - b^{2}}{2} = \frac{1}{2} [a^{2} + b^{2} - c^{2} - c^{2} - a^{2} + b^{2}] = \frac{1}{2} [2 b^{2} - 2 c^{2}] = b^{2} - c^{2}$

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