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The value of the series $\cos 12^{\circ}+\cos 84^{\circ}$ $+\cos 132^{\circ}+\cos 156^{\circ}$ is
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Verified Answer
The correct answer is:
$\frac{-1}{2}$
We have,
$$
\begin{aligned}
& \cos 12^{\circ}+\cos 84^{\circ}+\cos 132^{\circ}+\cos 156^{\circ} \\
= & \cos 132^{\circ}+\cos 12^{\circ}+\cos 156^{\circ}+\cos 84^{\circ} \\
= & 2 \cos 72^{\circ} \cdot \cos 60^{\circ}+2 \cos 120^{\circ} \cdot \cos 36^{\circ} \\
= & 2\left(\frac{\sqrt{5}-1}{4}\right) \frac{1}{2}+2\left(\frac{-1}{2}\right)\left(\frac{\sqrt{5}+1}{4}\right) \\
= & \frac{\sqrt{5}-1}{4}-\frac{\sqrt{5}+1}{4}=-\frac{2}{4} \\
= & -\frac{1}{2}
\end{aligned}
$$
$$
\begin{aligned}
& \cos 12^{\circ}+\cos 84^{\circ}+\cos 132^{\circ}+\cos 156^{\circ} \\
= & \cos 132^{\circ}+\cos 12^{\circ}+\cos 156^{\circ}+\cos 84^{\circ} \\
= & 2 \cos 72^{\circ} \cdot \cos 60^{\circ}+2 \cos 120^{\circ} \cdot \cos 36^{\circ} \\
= & 2\left(\frac{\sqrt{5}-1}{4}\right) \frac{1}{2}+2\left(\frac{-1}{2}\right)\left(\frac{\sqrt{5}+1}{4}\right) \\
= & \frac{\sqrt{5}-1}{4}-\frac{\sqrt{5}+1}{4}=-\frac{2}{4} \\
= & -\frac{1}{2}
\end{aligned}
$$
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