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  4. In Δ A B C,(a-b)2 cos 2 (C/2)+(a +b)2 sin 2 (C/2) is equal to

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Q. In $\Delta A B C,(a-b)^{2} \cos ^{2} \frac{C}{2}+(a +b)^{2} \sin ^{2} \frac{C}{2}$ is equal to

ManipalManipal 2008

Solution:

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