Question:
Mark the correct alternative in each of the following:
In any $\Delta \mathrm{ABC}, a(b \cos C-c \cos B)=$
(a) $a^{2}$
(b) $b^{2}-c^{2}$
(c) 0
(d) $b^{2}+c^{2}$
Solution:
Using cosine rule, we have
$a(b \cos C-c \cos B)$
$=a b\left(\frac{a^{2}+b^{2}-c^{2}}{2 a b}\right)-c a\left(\frac{c^{2}+a^{2}-b^{2}}{2 c a}\right)$
$=\frac{a^{2}+b^{2}-c^{2}-c^{2}-a^{2}+b^{2}}{2}$
$=\frac{2 b^{2}-2 c^{2}}{2}$
$=b^{2}-c^{2}$
Hence, the correct answer is option (b).