What's the prime period of the following function?
$\frac{\sin 2x + \cos 2x}{\sin 2x - \cos 2x}$
What's the prime period of the following function?
$\frac{\sin 2x + \cos 2x}{\sin 2x - \cos 2x}$
$\begin{align*} \frac{\sin 2x+\cos 2x}{\sin 2x-\cos 2x}&=\frac{\sin 2x+\cos 2x}{\sin 2x-\cos 2x}\cdot\frac{\sin 2x+\cos 2x}{\sin 2x+\cos 2x}\\\\ &=\frac{(\sin 2x+\cos 2x)^2}{\sin^2 2x-\cos^2 2x}\\\\ &=-\frac{\sin^2 2x+2\sin 2x\cos 2x+\cos^2 2x}{\cos 4x}\\\\ &=-\frac{1+\sin 4x}{\cos 4x}\\\\ &=-\sec 4x-\tan 4x\;. \end{align*}$
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?