# 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

## 1 Answer

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Good, but it will be even better if you additionally post a hint for the problem. To the extent possible, we try not to depend solely on external sites for our answers (to prevent link rot). –  Srivatsan Sep 12 '11 at 13:52
To be honest enough, that particular website is specifically oriented towards Mathematical Olympiads.One top member of this website also happens(was active there while till recently) to have an account there.(Qiaochu Yuan).Thanks for the suggestion though. –  Eisen Sep 12 '11 at 14:00
At the very least, merely supplying a link is best done in comments, not in answers. –  Ｊ. Ｍ. Sep 12 '11 at 14:33
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