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.
Best Answer
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).