Period of $f(x)=\sin{2x}-\sin{\frac{x}{2}}$ I know the periods of the two functions $f(x)=\sin{2x}$ and $g(x)=\sin{\frac{x}{2}}$, which are $\pi$ and $4\pi$ respectively. But what is their period when you subtract them? 
More generally, if $f(x)=\sin{(ax+c)}$ and $g(x)=\sin{(bx+d)},$ what is the period of $f(x)-g(x)?$ 

In short, period of $\sin 2x$ is $\pi$ and period of $\sin \dfrac{x}{2}$ is $4\pi$. Then answer will be $\text{lcm} (1,4)\cdot\pi=4\pi$

Now, elaborately:
$\sin 2x-\sin \dfrac{x}{2}=2\sin (x-\dfrac{x}{4})\cos(x+\dfrac{x}{4})=2\sin \dfrac{3x}{4}\cos\dfrac{5x}{4}$
If $x=4k\pi$ and $k\in \mathbb{Z}$, either $\cos\dfrac{5x}{4}$ or $\sin \dfrac{3x}{4}$ is $0$. Hence period is $4\pi$.
