cos 12^{circ} + cos 60^{circ} + cos 84^{circ} | vedclass

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Question

(C) Let $S = \cos 12^{\circ} + \cos 60^{\circ} + \cos 84^{\circ} + \cos 132^{\circ} + \cos 156^{\circ}$.We know that $\cos 60^{\circ} = \frac{1}{2}$.S

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(C) Let $S = \cos 12^{\circ} + \cos 60^{\circ} + \cos 84^{\circ} + \cos 132^{\circ} + \cos 156^{\circ}$.We know that $\cos 60^{\circ} = \frac{1}{2}$.So,$S = \frac{1}{2} + (\cos 12^{\circ} + \cos 84^{\circ} + \cos 132^{\circ} + \cos 156^{\circ})$.Using the sum-to-product formula $\cos A + \cos B = 2 \cos \frac{A+B}{2} \cos \frac{A-B}{2}$:$\cos 12^{\circ} + \cos 156^{\circ} = 2 \cos \frac{168^{\circ}}{2} \cos \frac{-144^{\circ}}{2} = 2 \cos 84^{\circ} \cos 72^{\circ}$.$\cos 84^{\circ} + \cos 132^{\circ} = 2 \cos \frac{216^{\circ}}{2} \cos \frac{-48^{\circ}}{2} = 2 \cos 108^{\circ} \cos 24^{\circ} = -2 \cos 72^{\circ} \cos 24^{\circ}$.This approach is complex. Let's use $\cos 132^{\circ} = \cos(180^{\circ}-48^{\circ}) = -\cos 48^{\circ}$ and $\cos 156^{\circ} = \cos(180^{\circ}-24^{\circ}) = -\cos 24^{\circ}$.$S = \frac{1}{2} + \cos 12^{\circ} + \cos 84^{\circ} - \cos 48^{\circ} - \cos 24^{\circ}$.$S = \frac{1}{2} + (\cos 84^{\circ} - \cos 48^{\circ}) + (\cos 12^{\circ} - \cos 24^{\circ})$.Using $\cos C - \cos D = -2 \sin \frac{C+D}{2} \sin \frac{C-D}{2}$:$\cos 84^{\circ} - \cos 48^{\circ} = -2 \sin 66^{\circ} \sin 18^{\circ}$.$\cos 12^{\circ} - \cos 24^{\circ} = -2 \sin 18^{\circ} \sin(-6^{\circ}) = 2 \sin 18^{\circ} \sin 6^{\circ}$.$S = \frac{1}{2} - 2 \sin 18^{\circ} (\sin 66^{\circ} - \sin 6^{\circ})$.Using $\sin C - \sin D = 2 \cos \frac{C+D}{2} \sin \frac{C-D}{2}$:$\sin 66^{\circ} - \sin 6^{\circ} = 2 \cos 36^{\circ} \sin 30^{\circ} = 2 \cos 36^{\circ} 	imes \frac{1}{2} = \cos 36^{\circ}$.$S = \frac{1}{2} - 2 \sin 18^{\circ} \cos 36^{\circ}$.Since $\sin 18^{\circ} = \frac{\sqrt{5}-1}{4}$ and $\cos 36^{\circ} = \frac{\sqrt{5}+1}{4}$:$S = \frac{1}{2} - 2 \left( \frac{\sqrt{5}-1}{4} ight) \left( \frac{\sqrt{5}+1}{4} ight) = \frac{1}{2} - 2 \left( \frac{5-1}{16} ight) = \frac{1}{2} - 2 \left( \frac{4}{16} ight) = \frac{1}{2} - \frac{1}{2} = 0$.

$\cos 12^{\circ} + \cos 60^{\circ} + \cos 84^{\circ} + \cos 132^{\circ} + \cos 156^{\circ} = $

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