The value of cos 12^circ + cos 84^circ + cos | vedclass

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(C) Given expression: $\cos 12^\circ + \cos 84^\circ + \cos 156^\circ + \cos 132^\circ$Group the terms: $(\cos 132^\circ + \cos 12^\circ) + (\cos 156^

Vedclass · Accepted Answer

$-\frac{1}{2}$

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(C) Given expression: $\cos 12^\circ + \cos 84^\circ + \cos 156^\circ + \cos 132^\circ$Group the terms: $(\cos 132^\circ + \cos 12^\circ) + (\cos 156^\circ + \cos 84^\circ)$Using the formula $\cos C + \cos D = 2 \cos \frac{C+D}{2} \cos \frac{C-D}{2}$:$= 2 \cos \frac{132^\circ + 12^\circ}{2} \cos \frac{132^\circ - 12^\circ}{2} + 2 \cos \frac{156^\circ + 84^\circ}{2} \cos \frac{156^\circ - 84^\circ}{2}$$= 2 \cos 72^\circ \cos 60^\circ + 2 \cos 120^\circ \cos 36^\circ$Since $\cos 60^\circ = \frac{1}{2}$ and $\cos 120^\circ = -\frac{1}{2}$:$= 2 \cos 72^\circ \left(\frac{1}{2}\right) + 2 \left(-\frac{1}{2}\right) \cos 36^\circ$$= \cos 72^\circ - \cos 36^\circ$Using values $\cos 72^\circ = \frac{\sqrt{5}-1}{4}$ and $\cos 36^\circ = \frac{\sqrt{5}+1}{4}$:$= \frac{\sqrt{5}-1}{4} - \frac{\sqrt{5}+1}{4} = \frac{\sqrt{5}-1-\sqrt{5}-1}{4} = \frac{-2}{4} = -\frac{1}{2}$.

The value of $\cos 12^\circ + \cos 84^\circ + \cos 156^\circ + \cos 132^\circ$ is

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