# What is the fundamental period of the function $f(x) = \sin x + \tan x + \tan\frac{x}{2} + \tan\frac{x}{4} + \tan\frac{x}{8} + \tan\frac{x}{16}$ .

by cmi   Last Updated August 14, 2019 06:20 AM

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 $$\pi$$.

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#### Answers 1

In the given function 16π is the fundamental period. The reason is if "T " is one of period then $$\frac{T}{n}$$ where n$${\into N }$$ 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.

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