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(D) Given expression: $\frac{\sin 3A - \cos \left( \frac{\pi}{2} - A \right)}{\cos A + \cos (\pi + 3A)}$Using trigonometric identities $\cos \left( \f

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$\cot 2A$

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(D) Given expression: $\frac{\sin 3A - \cos \left( \frac{\pi}{2} - A \right)}{\cos A + \cos (\pi + 3A)}$Using trigonometric identities $\cos \left( \frac{\pi}{2} - A \right) = \sin A$ and $\cos (\pi + 3A) = -\cos 3A$:$= \frac{\sin 3A - \sin A}{\cos A - \cos 3A}$Applying sum-to-product formulas $\sin C - \sin D = 2 \cos \left( \frac{C+D}{2} \right) \sin \left( \frac{C-D}{2} \right)$ and $\cos D - \cos C = 2 \sin \left( \frac{C+D}{2} \right) \sin \left( \frac{C-D}{2} \right)$:$= \frac{2 \cos \left( \frac{3A+A}{2} \right) \sin \left( \frac{3A-A}{2} \right)}{2 \sin \left( \frac{3A+A}{2} \right) \sin \left( \frac{3A-A}{2} \right)}$$= \frac{2 \cos 2A \sin A}{2 \sin 2A \sin A}$$= \frac{\cos 2A}{\sin 2A} = \cot 2A$

$\frac{\sin 3A - \cos \left( \frac{\pi}{2} - A \right)}{\cos A + \cos (\pi + 3A)} = $

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