<s>[INST] What's the prime period of $\frac{\sin 2x + \cos 2x}{\sin 2x - \cos 2x}$ What's the prime period of the following function?
$$\frac{\sin 2x + \cos 2x}{\sin 2x - \cos 2x}$$
 [/INST] $$\begin{align*}
\frac{\sin 2x+\cos 2x}{\sin 2x-\cos 2x}&=\frac{\sin 2x+\cos 2x}{\sin 2x-\cos 2x}\cdot\frac{\sin 2x+\cos 2x}{\sin 2x+\cos 2x}\\\\
&=\frac{(\sin 2x+\cos 2x)^2}{\sin^2 2x-\cos^2 2x}\\\\
&=-\frac{\sin^2 2x+2\sin 2x\cos 2x+\cos^2 2x}{\cos 4x}\\\\
&=-\frac{1+\sin 4x}{\cos 4x}\\\\
&=-\sec 4x-\tan 4x\;.
\end{align*}$$
The first term has primitive period $\dfrac{2\pi}4=\dfrac{\pi}2$, and the second has primitive period $\dfrac{\pi}4$. Can you finish it from there?
</s>