Why is $$\text{lcm}\left(\frac{\pi}{5},\frac{\pi}{2}\right)=\pi$$ where the $10$ here represents the period of $2\cos(10t+1)-\sin(4t-1)$
where $\displaystyle\frac{\pi}{5}$ is the period of $2\cos(10t+1)$ and $\displaystyle\frac{\pi}{2}$ is the period of $-\sin(4t-1)$.
The question is not why the lcm but it's why does the lcm of these two give $\pi$.
Suppose $n\pi/5=m\pi/2$ where $n,m\in\Bbb Z_{>0}$. For least multiple we need to minimize $m$. We get $n/m=5/2$ and the minimum $m$ that gives $n$ integral is $m=2$.