# Average and minimum Values of $|\sin x+ \cos x + \tan x + \cot x +\sec x +\csc x|$, $\forall x \in \mathbb{R}$

A problem was asked at Putnam Competition in 2003 (Problem 3), about finding the minimum Value of $|\sin x+ \cos x + \tan x + \cot x +\sec x +\csc x|$ where $x$ is Real.

I was thinking if there was any other simpler way to solve this problem. What strategy one should follow to determine the average value of above function?

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At the very least, you know that the function is $2\pi$-periodic, and is singular at $x=\pi/2$ within $(0,2\pi)$. – J. M. Sep 12 '11 at 12:47
Proceeding from @J.M.'s comments. As $x \to 0^+$, the function is lower bounded by $\csc x$, which in turn is at least $\frac{1}{x}$. Since the integral $\int_{0}^{a} \frac{1}{x}$ diverges for any $a > 0$, the average value of this function, over say $[0,2\pi]$, is also infinite. – Srivatsan Sep 12 '11 at 15:27

That is definitely a good read (pretty much related to 'solutions given on AMC.MAA.ORG site'), but what about the average value of $|\sin x+ \cos x + \tan x + \cot x +\sec x +\csc x|$ ? I have not even an Idea to how to start working on this! – gaurav Sep 12 '11 at 14:49