What is the fundamental period of the function $ f(x) = \sin x + \tan x + \tan\frac{x}{2} + \tan\frac{x}{4} + \tan\frac{x}{8} + \tan\frac{x}{16}$ . What is the fundamental period of the function $$ f(x) = \sin x + \tan x + \tan\frac{x}{2} + \tan\frac{x}{4} + \tan\frac{x}{8}  + \tan\frac{x}{16}~ .$$
I know that $16\pi$ is one period but how can I determine the fundamental period? 
Can anyone please help me to find out it's fundamental period? 

My friend was telling me that it's fundamental period will also be $~16\pi~$. Because $~16\pi~$ is the L.C.M of all periods of the periodic functions in the expression. 
But I can not understand this argument because the well known function $~|\sin x | + |\cos x|~$ is a periodic function with period $~\frac{\pi}{2}~$ where as $~|\sin x |~$  and  $~|\cos x|~$ are of period $~\pi  ~$.
 A: It is easy to check that $f(x + 16\pi) = f(x)$ for all $x \in \mathbb{R}$. Since $f$ is continuous on its domain (or even continuous as function $\mathbb{R} \to  \mathbb{R}\cup\{\infty\}$) and non-constant, it follows that the fundamental period of $f$ is of the form $16\pi/n$ for some positive integer $n$.
Now if $n$ is a positive integer for which $16\pi/n$ is a period of $f$, then we must have $f(16\pi/n) = 0$.


*

*If $n > 32$, then it is easy to see that each summand of $f(16\pi/n)$ is positive, and so, $f(16\pi/n) > 0$.

*So it suffices to check that $f(16\pi/n) \neq 0$ for each $2 \leq n \leq 32$. This is the trickiest part, and to be honest, I do not see any clear argument for this. (Although we can easily remove $n = 2, 4, 8, 16, 32$ out of options, all the other values still deserve to be investigated.) But any CAS is capable of computing those values, and it turns that none of them are zero.

*Therefore the only possible choice is $16\pi$.
A: In the given function 16π is the fundamental period. The reason is if "T " is one of period then $ \frac{T}{n} $ where n$ {\in N } $ can be a fundamental period if $\\ f(x\,+\,T)\,=f(x) $ . Generally it happens in function which transform from one to the other given e.g |Sin((π/2)+x)| = | Cos x | and vise versa there fundamental period get refused to π/2.
A: If $T_1,T_2,T_3,...$ are the periods of individually added functions and if the ratios of these periods are rational the period of the the whole function is given by the $LCM(T_1,T_2,T_3...)$ If $T$ is period of $g(x)$, the period of $g(ax)$ is $T/a$. So the period of the whole given function is
$$T=LCM(2\pi,\pi,2\pi, 4\pi, 8\pi, 16 \pi)=16 \pi$$
