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

The period of $~\sin x~$ is $~2\pi~$ and  that of $~\sin 3x~$ is $~\frac{2\pi}{3}~$, because $~\sin\left(3\frac{2\pi}{3}\right)=\sin 2\pi~$.
Now given that $~f(x)=\sin 3x+\sin x~$
Let $~a~$be a period $~f(x)~$, then by the definition $~f(x+a)=f(x)~$.
Here $~f(x+a)=\sin(3x+3a)+\sin(x+a)~$
From $~\sin(3x+3a)~$, we have $~a=\frac{2\pi}{3}n,~~ n$ is integer, because its the same as adding period of $~\sin 3x, ~~n$ times.
Similarly, from $~\sin(x+a)~$, we have $~a=2\pi k~$, $k$ is integer.
$~a~$ is a multiple of $~\frac{2\pi}{3}~$ and $~2\pi~$, so the period is smallest positive multiple of $~\frac{2\pi}{3}~$ and $~2\pi~$ which is $~2\pi~$, because $~2\pi=2\pi*1 (\text{multiple of}~ 2\pi), 2\pi=\frac{2\pi}{3}*3 (\text{multiple of}~ \frac{2\pi}{3})~$.
The period of $~f(x)~$, $~a=2\pi~$
