What is the period of the $f(x)=\sin x +\sin3x$? 
What is the period of the $f(x)=\sin x +\sin3x$?

$f(x)=\sin x+\sin 3x=2\frac{3x+x}{2}\cos\frac{x-3x}{2}=2\sin2x\cos x=4\sin x\cos^2x\\f(x+T)=4\sin(x+T)\cos^2(x+T)=4\sin x\cos^2 x$
how can I deduct this I have no idea

In general, if $T$ is the period of a function $f(x)$ then the period of the function $f(ax)$ is $\frac{T}{a}$.
Suppose two periodic functions $f_1(x)$ and  $f_2(x)$ have periods $T_1$ and $T_2$. Then the period of the function $g(x)=f_1(x)\pm f_2(x)$ is LCM (least common multiple) of $T_1$ and $T_2$ (although, this certainly isn't true for all periodic functions, as explained inside this answer.)
In your question the periods of $\sin x$ and $\sin 3x$ are calculated as $\frac{2\pi}{1}=2\pi$ and $\frac{2\pi}{3}$ respectively. 
So the period of the function $f(x)=\sin x+\sin3x$ is the $\text{LCM}(2\pi,\frac{2\pi}{3})=2\pi$.
