Question:
Prove that
$\cos x+\cos \left(120^{\circ}-x\right)+\cos \left(120^{\circ}+x\right)=0$
Solution:
In this question the following formulas will be used:
$\cos (A+B)=\cos A \cos B-\sin A \sin B$
$\cos (A-B)=\cos A \cos B+\sin A \sin B$
$=\cos x+\cos 120^{\circ} \cos x-\sin 120 \sin x+\cos 120^{\circ} \cos x+\sin 120 \sin x$
$=\cos x+2 \cos 120 \cos x$
$=\cos x+2 \cos (90+30) \cos x$
$=\cos x+2(-\sin 30) \cos x$
$=\cos x-2 \times \frac{1}{2} \cos x$
$=\cos x-\cos x$
$=0$.
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