Question

The period of function $f\left(x\right)=\sin \frac{x}{n!}+\cos \frac{x}{(n+1)!}$ is -

Answer 1

$\sin \left(\frac{x}{n!}\right)+\cos \left(\frac{x}{(n+1)!}\right)$

put $\frac{x}{n!}=\left(\theta \right)$

$\Rightarrow \sin \left(\theta \right)+\cos \left(\frac{\theta }{n+1}\right)$

period of $\sin \left(\theta \right)=2\pi$

period of $\cos \left(\frac{\theta }{n+1}\right)=2\pi \cdot (n+1)$

$\Rightarrow$ period of $\sin \theta +\cos \frac{\theta }{n+1}=(n+1)2\pi$

$\Rightarrow$ period of $\sin \frac{x}{n!}+\cos \frac{x}{(n+1)!}$

$=(n+1)\cdot 2\pi \cdot n!$