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What is the value of $\cos 10^{\circ}+\cos 110^{\circ}+\cos 130^{\circ} ?$
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Consider $\cos 10^{\circ}+\cos 110^{\circ}+\cos 130^{\circ}$
$=\cos 130^{\circ}+\cos 10^{\circ}+\cos 110^{\circ}$
$=2 \cos \left(\frac{130+10}{2}\right) \cos \left(\frac{130-10}{2}\right)+\cos 110^{\circ}$
$=2 \cos \left(\frac{140}{2}\right) \cos \left(\frac{120}{2}\right)+\cos 110^{\circ}$
$=2 \cos 60^{\circ} \cos 70^{\circ}+\cos 110^{\circ}$
$=\cos 70^{\circ}+\cos 110^{\circ} \quad\left(\because \cos 60^{\circ}=\frac{1}{2}\right)$
$=\cos \left(180^{\circ}-110^{\circ}\right)+\cos 110^{\circ}$
$=-\cos 110^{\circ}+\cos 110^{\circ}=0\left(\because \cos \left(180^{\circ}-\theta\right)=-\cos \theta\right)$
$=\cos 130^{\circ}+\cos 10^{\circ}+\cos 110^{\circ}$
$=2 \cos \left(\frac{130+10}{2}\right) \cos \left(\frac{130-10}{2}\right)+\cos 110^{\circ}$
$=2 \cos \left(\frac{140}{2}\right) \cos \left(\frac{120}{2}\right)+\cos 110^{\circ}$
$=2 \cos 60^{\circ} \cos 70^{\circ}+\cos 110^{\circ}$
$=\cos 70^{\circ}+\cos 110^{\circ} \quad\left(\because \cos 60^{\circ}=\frac{1}{2}\right)$
$=\cos \left(180^{\circ}-110^{\circ}\right)+\cos 110^{\circ}$
$=-\cos 110^{\circ}+\cos 110^{\circ}=0\left(\because \cos \left(180^{\circ}-\theta\right)=-\cos \theta\right)$
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