# How to find the minimum of $f(x)=(\sin(x)+\cos(x)+\tan(x)+\cot(x)+\sec(x)+\csc(x))^2$?

I need to find the minimum of $f(x)$ with $$f(x)=(\sin(x)+\cos(x)+\tan(x)+\cot(x)+\sec(x)+\csc(x))^2$$ Could you help me with some clues?

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Do you know what a derivative is? – ja72 Apr 13 '13 at 20:41
@ja72 yes, I know. – Queayiouer Apr 13 '13 at 20:45
I found the answer numerically, but I am not sure how to proceed analytically. – ja72 Apr 13 '13 at 20:48
Calculating the derivative will not work. – Lord Soth Apr 13 '13 at 20:49
@LordSoth it's true. Calculating the derivative the difficulty remains unchanged. – Queayiouer Apr 13 '13 at 20:51

You can do the following:

Let $$\sin x + \cos x = y$$

Then we have that $$\tan x + \cot x = \frac{2}{y^2 -1}$$

and

$$\sec x + \csc x = \frac{2y}{y^2-1}$$

I believe $$\sin x + \cos x + \tan x + \cot x + \sec x + \csc x$$ simplifies to $$y + \frac{2}{y-1}$$ (but I tried doing it in my head, so might have made mistakes).

And you can use the standard calculus techniques, now. (but take care to eliminate the corner cases etc).

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I can do it from here. Thank you. – Queayiouer Apr 13 '13 at 21:09
(+1) This is classic.:) My answer hint no where.. – Inceptio Apr 13 '13 at 21:11

To built upon @Queayiouer if you split the function as

$$f(x) = \left( g(x) \right)^2 = \left( \left(\sin x+\frac{1}{\sin x}\right) + \left(\cos x+\frac{1}{\cos x}\right) + \left(\tan x+\frac{1}{\tan x}\right) \right)^2$$

where the minimum occurs if $g(x)=0$ or $g'(x)=0$. The first is not going to happen.

So now we have

$$g(x) = g_1(x) + g_2(x) + g_3(x) = \left(\sin x+\frac{1}{\sin x}\right) + \left(\cos x+\frac{1}{\cos x}\right) + \left(\tan x+\frac{1}{\tan x}\right)$$

Let find if they three functions have a common minimum since

$$g_1'(x) = \cos(x) \left(1-\frac{1}{\sin^2 x} \right) = 0$$ $$g_2'(x) = \sin(x) \left(\frac{1}{\cos^2 x}-1 \right) = 0$$ $$g_3'(x) = \tan^2 x - \frac{1}{\tan^2 x} = 0$$

Unfortunately those are never all zero at the same time. So my guess it to proceed numerically with Newton-Raphson method where

$$x \rightarrow x - \frac{ g_1(x) + g_2(x) + g_3(x) }{ g_1'(x) + g_2'(x) + g_3'(x) }$$

starting from $x=1$ or something. I get between $2.5<x<2.7$.

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