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Question
Write the value of \[\frac{\sin A + \sin 3A}{\cos A + \cos 3A}\]
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Solution
\[\frac{\sin A + \sin3A}{\cos A + \cos3A}\]
\[ = \frac{2\sin\left( \frac{A + 3A}{2} \right)\cos\left( \frac{A - 3A}{2} \right)}{2\cos\left( \frac{A + 3A}{2} \right)\cos\left( \frac{A - 3A}{2} \right)} \left[ \because \sin A + \sin B = 2\sin\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right), \text{ and }\cos A + \cos B = 2\cos\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right) \right]\]
\[ = \frac{\sin2A \cos\left( - A \right)}{\cos2A \cos\left( - A \right)}\]
\[ = \frac{\sin2A \cos A}{\cos2A \cos A}\]
\[ =\tan2A\]
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