Determine the period of,
$$g(x)=\cos2x+\cos(\frac{2x}{3})+\cos(\frac{2x}{5})$$
My thinking is that if three periodic functions $f_1(x), f_2(x), f_3(x)$ have periods $T_1, T_2, T_3$ then the period of the function $g(x)=f_1(x)\pm f_2(x) \pm f_3(x)$ is L.C.M. (Least common multiple) of $T_1, T_2, T_3$.
Using the equation that $P = \frac{2\pi}{B}$
So $f_1(x) = \cos2x$ then $f_1(x) = 2 \cos x$, then the period of that would be $\pi$
So $f_2(x) = \cos(\frac{2x}{3})$, then $f_2(x) = \frac{2}{3}\cos x$ , then the period would be $3\pi$
So $f_3(x) = \cos(\frac{2x}{5})$, then $f_3(x) = \frac{2}{5}\cos x$, then the period would be $5\pi$
So then the LCM of $f_1(x), f_2(x), f_3(x)$ would be $15\pi$.
I am not sure if this method is correct or if there is another method to use like taking,
$\cos(2x)=\cos(2(x+k\pi)) = \pi $
Your method is correct. Alternatively: $$g(x+T)=\cos{(2x+2T)}+\cos{(\frac{2x}{3}+\frac{2T}{3})}+\cos{(\frac{2x}{5}+\frac{2T}{5})}=g(x),$$ if $T=15\pi$.