cos frac{2 pi}{7}+cos frac{4 pi}{7}+cos frac{ | vedclass

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(C) Let $S = \cos \frac{2 \pi}{7} + \cos \frac{4 \pi}{7} + \cos \frac{6 \pi}{7}$.Multiply by $2 \sin \frac{\pi}{7}$:$2 \sin \frac{\pi}{7} S = 2 \sin \

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is a negative number

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(C) Let $S = \cos \frac{2 \pi}{7} + \cos \frac{4 \pi}{7} + \cos \frac{6 \pi}{7}$.Multiply by $2 \sin \frac{\pi}{7}$:$2 \sin \frac{\pi}{7} S = 2 \sin \frac{\pi}{7} \cos \frac{2 \pi}{7} + 2 \sin \frac{\pi}{7} \cos \frac{4 \pi}{7} + 2 \sin \frac{\pi}{7} \cos \frac{6 \pi}{7}$.Using $2 \sin A \cos B = \sin(A+B) - \sin(B-A)$:$2 \sin \frac{\pi}{7} S = (\sin \frac{3 \pi}{7} - \sin \frac{\pi}{7}) + (\sin \frac{5 \pi}{7} - \sin \frac{3 \pi}{7}) + (\sin \frac{7 \pi}{7} - \sin \frac{5 \pi}{7})$.All terms cancel out except $\sin \frac{7 \pi}{7} - \sin \frac{\pi}{7}$.Since $\sin \pi = 0$, we have $2 \sin \frac{\pi}{7} S = -\sin \frac{\pi}{7}$.Since $\sin \frac{\pi}{7} 
eq 0$, we get $S = -\frac{1}{2}$.Thus, the sum is a negative number.

List-$A$	List-$B$
$(I)$ $\tan \left(\frac{\alpha + \beta}{2}\right) =$	$(a)$ $\frac{b}{a}$
$(II)$ $\cos (\alpha + \beta) =$	$(b)$ $\frac{2ab}{a^2 + b^2}$
$(III)$ $\sin (\alpha + \beta) =$	$(c)$ $\frac{2ab}{a^2 - b^2}$
$(IV)$ $\tan (\alpha + \beta) =$	$(d)$ $\frac{a^2 - b^2}{a^2 + b^2}$

$\cos \frac{2 \pi}{7}+\cos \frac{4 \pi}{7}+\cos \frac{6 \pi}{7}$

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