$\text{ What is the fundamental period of }f\left(x\right)=\frac{\sin x+\sin 3x}{\cos x+\cos 3x}\text{ ? }$

$$\begin{gathered} f\left(x\right) =\frac{\sin x+\sin 3x}{\cos x+\cos 3x} \\ \Rightarrow f\left(\mathrm{\pi } +\mathrm{x}\right)=\frac{\sin \left(\mathrm{\pi }+\mathrm{x}\right) +\sin 3\left(\mathrm{\pi }+\mathrm{x}\right)}{\cos \left(\mathrm{\pi }+\mathrm{x}\right)+\cos 3\left(\mathrm{\pi }+\mathrm{x}\right)} =\frac{-\sin x-\sin 3x}{-\cos x-\cos 3x}=f\left(x\right) \end{gathered}$$

$Thus f\left(x \right) period is \mathrm{\pi } . \left(\mathrm{since} \mathrm{f}\left(\frac{\mathrm{\pi }}{2}+\mathrm{x}\right) \ne \mathrm{f}\left(\mathrm{x}\right)\right)$