Prove that


Prove that

$\sin \left(150^{\circ}+x\right)+\sin \left(150^{\circ}-x\right)=\cos x$



In this question the following formula will be used:

$\sin (A+B)=\sin A \cos B+\cos A \sin B$

$\sin (A-B)=\sin A \cos B-\cos A \sin B$

$=\sin 150^{\circ} \cos x+\cos 150^{\circ} \sin x+\sin 150^{\circ} \cos x-\cos 150^{\circ} \sin x$

$=2 \sin 150^{\circ} \cos x$

$=2 \sin \left(90^{\circ}+60^{\circ}\right) \cos x$

$=2 \cos 60^{\circ} \cos x$

$=2 \times \frac{1}{2} \cos x$

$=\cos x$


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