The period of the function $f(x) = \frac{2 \sin \left(\frac{\pi x}{3}\right) \cos \left(\frac{2 \pi x}{5}\right)}{3 \tan \left(\frac{7 \pi x}{2}\right) - 5 \sec \left(\frac{5 \pi x}{3}\right)}$ is

If the period of the function $f(x) = 2 \cos(3x + 4) - 3 \tan(2x - 3) + 5 \sin(5x) - 7$ is $k$,then

The period of the function $f(x) = e^{x - [x] + |\cos \pi x| + |\cos 2\pi x| + \dots + |\cos n\pi x|}$ (where $[.]$ denotes the greatest integer function) is:

Choose the correct statement:

The period of $\frac{\sin \theta + \sin 2\theta}{\cos \theta + \cos 2\theta}$ 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} = $