સાબિત કરો કે : $\frac{\sin 5 x+\sin 3 x}{\cos 5 x+\cos 3 x}=\tan 4 x$

It is known that

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

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

$\therefore$ $L.H.S.$ $=\frac{\sin 5 x+\sin 3 x}{\cos 5 x+\cos 3 x}$

$=\frac{2 \sin \left(\frac{5 x+3 x}{2}\right) \cdot \cos \left(\frac{5 x-3 x}{2}\right)}{2 \cos \left(\frac{5 x+3 x}{2}\right) \cdot \cos \left(\frac{5 x-3 x}{2}\right)}$

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

$=\tan 4 x=R . H . S$