frac{{sin 3A - cos left( {frac{pi }{2} - A} r | vedclass

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Question

(D) Given expression: $\frac{{\sin 3A - \cos \left( {\frac{\pi }{2} - A} \right)}}{{\cos A + \cos (\pi + 3A)}}$Using trigonometric identities:$1$. $\c

Vedclass · Accepted Answer

$\cot 2A$

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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:$1$. $\cos \left( {\frac{\pi }{2} - A} \right) = \sin A$$2$. $\cos (\pi + 3A) = -\cos 3A$Substituting these into the expression:$= \frac{{\sin 3A - \sin A}}{{\cos A - \cos 3A}}$Applying sum-to-product formulas:$\sin C - \sin D = 2 \cos \left( \frac{C+D}{2} \right) \sin \left( \frac{C-D}{2} \right)$$\cos D - \cos C = 2 \sin \left( \frac{C+D}{2} \right) \sin \left( \frac{C-D}{2} \right)$Numerator: $\sin 3A - \sin A = 2 \cos 2A \sin A$Denominator: $\cos A - \cos 3A = 2 \sin 2A \sin A$$= \frac{{2 \cos 2A \sin A}}{{2 \sin 2A \sin A}}$$= \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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