Main period of $f(x)=\cos 5x+\cos 10x$ Here is what I have done so far:
\begin{align*}
f(x)&=\cos5x+\cos10x\\
f(x)&=\cos5x+2\cos^2(5x)-1\\
f(x)&=2\cos^25x+\cos5x-1\\
\end{align*}
I have tried to further simplify the function to a complete square or a function like $\cos^2(\text{something})$ but I was not able to. Then I factorised:
$$f(x)=2\left(\cos5x+\frac12\right)(\cos5x-1),$$
where each multiplier has a period of $\dfrac{2π}{5}$, which makes me think that the main period of the function is $\dfrac{2π}{5}$. 
Please excuse me for the bad terminology. English is not my native language.
 A: You don't have to simply anything.
Hint:
Time period of $\displaystyle \cos (ax)=\frac{2\pi}{a}$
Also, the time period of sum of two functions $f_1(x)$ and $f_2(x)$ with period $T_1$ and $T_2$ is $\mathrm{LCM}(T_1$, $T_2)$
Also note that: $$\mathrm{LCM}\left(\frac{p_1}{q_1},\frac{p_2}{q_2}\right)=\frac{\mathrm{LCM}(p_1,p_2)}{\mathrm{HCF}(q_1,q_2)}$$
As a side note, before finding the time period of the sum always check if the sum is periodic or not. You can do that by checking if $\frac{T_1}{T_2}$ is rational (periodic) or not (aperiodic).
A: To calculate the period for any function of the form: 
$$f(x) = a(x) + b(x) + c(x)+ \cdots$$
At first, find the periods of individual functions $a(x), b(x), c(x), \ldots$
The period of $f(x)$ would then be the least common multiples of the periods of $a(x), b(x), c(x), \ldots$
The logic makes sense, since if $a(x)$ repeats in $t_1$ period, $b(x)$ repeats in t2 period, $c(x)$ repeats in $t_3$ period and so on, the whole function will repeat in multiples of $t_1, t_2, t_3$ and so on. Since period is the least interval where the function repeats, we calculate the least common multiple of all the multiples of $t_1, t_2, t_3$ and so on, to find the period of $f(x)$.
In this case, period of $\sin(5x)$ is $\frac{2\pi}{5}$, and for $\sin(10x)$ is $\frac{2\pi}{10}$. The lest common multiples of these two intervals is $\frac{2\pi}{5}$, which is the period of $f(x)$. Thanks.
