<s>[INST] Determining the period of $\sin 2x +\sin\frac{x}{2}$ without using the LCM of periods 
I tried to calculate period of function described as: 
  $$y=\sin 2x +\sin\frac{x}{2}$$ 
  but without using LCM of periods.

From definition of periodic function we have:
$$\begin{align}
0 &= \sin(2x+2T)+\sin\left(\frac{x+t}{2}\right)-\sin2x-\sin\left(\frac{x}{2}\right) \\[4pt]
&=2\sin\left(\frac{5x+5T}{4}\right)\cos\left(\frac{x+T}{4}\right)-2\sin\left(\frac{5x}{4}\right)\cos\left(\frac{3x}{4}\right)
\end{align}$$
I cannot do further more.
 [/INST] Let $$\sin2(x+T)+\sin\frac{x+T}{2}=\sin2x+\sin\frac{x}{2},$$ where $T>0$.
Thus, for $x=0$ we obtain:
$$\sin2T+\sin\frac{T}{2}=0$$ and for $x=2\pi$ we obtain:
$$\sin2T-\sin\frac{T}{2}=0,$$
which gives $$\sin2T=\sin\frac{T}{2}=0$$ or
$$T=2\pi k,$$ where $k$ is a positive integer number.
We see that $k=1$ is not valid, but $k=2$ gives the answer:
$$T=4\pi.$$
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