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Simplify: cos(30° - A) - cos(30° + A)
Question

Simplify: cos(30° - A) - cos(30° + A)

A.

cos A

B.

(1/2) sin A

C.

sin A

D.

2sinA

Correct option is C

Given: ​
​cos(30° - A) - cos(30° + A) 
​Formula Used:cos⁡(x−y)=cos⁡xcos⁡y+sin⁡xsin⁡ycos⁡(x+y)=cos⁡xcos⁡y−sin⁡xsin⁡ySolution:cos⁡(30∘−A)=cos⁡30∘cos⁡A+sin⁡30∘sin⁡Acos⁡(30∘+A)=cos⁡30∘cos⁡A−sin⁡30∘sin⁡AThus solving,cos⁡(30∘−A)−cos⁡(30∘+A)=cos⁡30∘×cos⁡A+sin⁡30∘×sin⁡A−cos⁡30∘×cos⁡A+sin⁡30∘×sin⁡Acos⁡(30∘−A)−cos⁡(30∘+A)=2×sin⁡30∘×sin⁡Acos⁡(30∘−A)−cos⁡(30∘+A)=2×12×sin⁡Acos⁡(30∘−A)−cos⁡(30∘+A)=sin⁡A\textbf{Formula Used:}\\\cos(x - y) = \cos x \cos y + \sin x \sin y \\\cos(x + y) = \cos x \cos y - \sin x \sin y \\\textbf{Solution}: \\\cos(30^\circ - A) = \cos 30^\circ \cos A + \sin 30^\circ \sin A \\\cos(30^\circ + A) = \cos 30^\circ \cos A - \sin 30^\circ \sin A \\\\\text{Thus solving,} \\\\\cos(30^\circ - A) - \cos(30^\circ + A) = \cos 30^\circ \times \cos A + \sin 30^\circ \times \sin A - \cos 30^\circ \times \cos A + \sin 30^\circ \times \sin A \\\cos(30^\circ - A) - \cos(30^\circ + A) = 2 \times \sin 30^\circ \times \sin A \\\cos(30^\circ - A) - \cos(30^\circ + A) = 2 \times \frac{1}{2} \times \sin A \\\cos(30^\circ - A) - \cos(30^\circ + A) = \sin A​​














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