The period of $\cot 3x - \cos (4x + 3)$ is

If $f(x) = \sin^2\left(\frac{\pi}{8} + \frac{x}{2}\right) - \sin^2\left(\frac{\pi}{8} - \frac{x}{2}\right)$,then the period of $f$ is

The period of the function $\sin \left( \frac{\pi x}{2} \right) + \cos \left( \frac{\pi x}{2} \right)$ is

The period of $\sin \frac{x}{2} - \cos \frac{x}{3}$ is (in $\pi$)

The period of $\frac{\sin x}{\cos 3x} + \frac{\sin 3x}{\cos 9x} + \frac{\sin 9x}{\cos 27x} + \frac{\sin 27x}{\cos 81x}$ is

Let $\alpha$ be the period of $3 \sin \frac{\pi x}{3} - \cos \frac{\pi x}{2} + \tan \frac{\pi x}{4}$,$\beta$ be the period of $\sin^2 \left( \frac{\pi}{7} + \frac{x}{4} \right) - \sin^2 \left( \frac{\pi}{7} - \frac{x}{4} \right)$,and $\gamma$ be the period of $\cos^4 x + \sin^4 x$. Then $\frac{\alpha \gamma}{\beta} = $