If $n \in N$ and the period of $\frac{\cos nx}{\sin \left(\frac{x}{n}\right)}$ is $4\pi$,then $n$ is equal to

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} = $

The period of $f(x) = \sin \left( \frac{\pi x}{n - 1} \right) + \cos \left( \frac{\pi x}{n} \right)$,where $n \in \mathbb{Z}$ and $n > 2$,is:

Let $f(x) = \cos px + \sin x$ be a periodic function,then $p$ must be

The period of $\tan(ky) + \sin(ky)$,where $k = 1 + 4 + 9 + \ldots$ ($20$ terms),is

Let $f$ and $g$ be periodic functions with the periods $T_{1}$ and $T_{2}$ respectively. Then $f+g$ is