Write the value of

$\frac{\sin A+\sin 3 A}{\cos A+\cos 3 A}$

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

$\left[\because \sin A+\sin B=2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)\right.$, and $\left.\cos A+\cos B=2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)\right]$

$=\frac{\sin 2 A \cos (-A)}{\cos 2 A \cos (-A)}$

$=\frac{\sin 2 A \cos A}{\cos 2 A \cos A}$

$=\tan 2 A$