Prove that

$\frac{\sin 5 x+\sin 3 x}{\cos 5 x+\cos 3 x}$

$=\frac{2 \sin \frac{5 x+3 x}{2} \cos \frac{5 x-3 x}{2}}{2 \cos \frac{5 x^{2}+3 x}{2} \cos \frac{5 x-3 x}{2}}$

$=\frac{2 \sin 4 x \cos x}{2 \cos 4 x \cos x}$

$=\tan 4 x$

Using the formula,

$\sin A+\sin B=2 \sin \frac{A+B}{2} \cos \frac{A-B}{2}$

$\cos A+\cos B=2 \cos \frac{A+B}{2} \cos \frac{A-B}{2}$