Simplify: $$\cos(30^\circ - A) - \cos(30^\circ + A)$$

$$ \cos (x-y) =\cos x \cos y +\sin x \sin y $$

$$ \cos (x+y) =\cos x \cos y -\sin x \sin y $$

$$ \cos (30-A) =\cos 30 \cos A+\sin 30 \sin A $$

$$ \cos (30+A) =\cos 30 \cos A -\sin 30 \sin A $$

Thus solving,

$$ \cos (30°-A) - \cos (30°+A) =\cos 30°×\cos A + \sin 30°×\sin A - \cos 30°×\cos A + \sin 30°×\sin A $$

 $$ \cos (30°-A) - \cos (30°+A) =2×\sin 30°×\sin A $$

$$ \cos (30°-A) - \cos (30°+A) =2×\frac{1}{2}×\sin A $$

$$ \cos (30°-A) - \cos (30°+A) =\sin A $$