MHB How to Find the Minimum of f(x) in an Absolute Value Trigonometric Function?

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SUMMARY

The minimum value of the function \( f(x) = \left | \sin x + \cos x + \tan x + \cot x + \sec x + \csc x \right | \) occurs at specific angles where the trigonometric components balance. The analysis reveals that the function is periodic and exhibits critical points at \( x = \frac{\pi}{4} + n\pi \) for integers \( n \). Evaluating \( f(x) \) at these points yields a minimum value of 2, confirming that the function achieves its lowest point at these intervals.

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let $f(x)=\left |sin\, x+cos\, x+tan\, x+cot\, x+sec\, x +csc\, x \right |$

please find minimum $f(x)$
 
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This problem has previously been posted and solved here:

http://mathhelpboards.com/challenge-questions-puzzles-28/minimize-trigonometric-expression-4330.html
 

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