We know minimum value of $\sin(x)+\cos(x)+\tan(x)+\cot(x)+\sec(x)+\csc(x)=6$ by AM-GM inequality. But I wanted to know whether manually can find out that angle $x$. Is it possible?
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1Equality holds when all $6$ of them are equal, and I doubt if that would happen. Also all $6$ of them have to be positive to use AM-GM. – peterwhy Apr 28 '16 at 16:33
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Yes that's what I want to solve $...=..=..$ – Archis Welankar Apr 28 '16 at 16:36
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If $\sin x = \csc x$, then $|\sin x| = 1$, but $\cos x = 0$ at those values, and $\sec x$ is not even defined. So $6$ is not a tight lower bound of the sum, assuming you are talking about $x\in(0,\pi/2)$. – peterwhy Apr 28 '16 at 16:40
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1The minimum is slightly bigger than 6. – AugSB Apr 28 '16 at 16:41
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No AugSB just wanted to know if its posibble let minimum value be $6...$ – Archis Welankar Apr 28 '16 at 16:54
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See http://math.stackexchange.com/questions/301800/find-the-minimum-value-of-sin-x-cos-x-tan-x-cot-x-sec-x-csc-x-for-real – lab bhattacharjee Apr 29 '16 at 04:45