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$A+B=C \quad \Rightarrow$
$\cos ^2 A+\cos ^2 B+\cos ^2 C-2 \cos A \cos B \cos C$ is equal to
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$\cos ^2 A+\cos ^2 B+\cos ^2 C-2 \cos A \cos B \cos C$ is equal to
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The correct answer is:
1
Given that, $A+B=C$
$\begin{aligned}
& \text { Now, } \cos ^2 A+\cos ^2 B+\cos ^2 C \\
& =\frac{1+\cos 2 A}{2}+\frac{1+\cos 2 B}{2}+\cos ^2 C \\
& =1+\frac{1}{2}(\cos 2 A+\cos 2 B)+\cos ^2 C \\
& =1+\frac{2}{2}[\cos (A+B) \cos (A-B)]+\cos ^2 C \\
& =1+\cos C[\cos (A-B)+\cos (A+B)] \\
& =1+2 \cos C \cos B \cos A \\
& \Rightarrow \cos ^2 A+\cos ^2 B+\cos ^2 C-2 \cos A \cos B \cos C=1
\end{aligned}$
$\begin{aligned}
& \text { Now, } \cos ^2 A+\cos ^2 B+\cos ^2 C \\
& =\frac{1+\cos 2 A}{2}+\frac{1+\cos 2 B}{2}+\cos ^2 C \\
& =1+\frac{1}{2}(\cos 2 A+\cos 2 B)+\cos ^2 C \\
& =1+\frac{2}{2}[\cos (A+B) \cos (A-B)]+\cos ^2 C \\
& =1+\cos C[\cos (A-B)+\cos (A+B)] \\
& =1+2 \cos C \cos B \cos A \\
& \Rightarrow \cos ^2 A+\cos ^2 B+\cos ^2 C-2 \cos A \cos B \cos C=1
\end{aligned}$
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