Q:

Minimum value of : $\left|\sin{x}+ \cos{x} + \sec{x} + \tan{x} + \cot{x} + \csc{x}\right|$

Find the minimum value of :$$\left|\sin{x}+ \cos{x} + \sec{x} + \tan{x} + \cot{x} + \csc{x}\right|.$$

What I have tried is trying to simplify the expression into a $\sin{2x}$ form but it did not go well. I guess there is much simpler way to solve the problem.
I also thought of AM-GM inequality but for inequality to hold all the terms must be positive real numbers which in this case is not.

A:

$$\bigg|\sin x+\cos x+\tan x+\cot x +\csc x+\sec x\bigg|$$
$$ = \bigg|\sin x+\cos x+\frac{1}{\sin x\cos x}+\frac{\sin x+\cos x}{\sin x\cos  x}\bigg|$$
$$ = \bigg|(\sin x+\cos x)+\frac{2}{\sin 2x}+\frac{2(\sin x+\cos x)}{\sin 2x}\bigg|$$
put $\sin x+\cos = t,$ and $\sin 2x = t^2-1$ where $|t|<=\sqrt{2}$
so $$\bigg|t+\frac{2}{t^2-1}+\frac{2t}{t^2-1}\bigg| = \bigg|t-1+\frac{2}{t-1}+1\bigg|$$
let $$y = \bigg(t-1+\frac{2}{t-1}+1\bigg)$$
$\star$ If $ 1<t <=\sqrt{2}. $ Then using A.M G.M , we have $$\displaystyle y=t-1+\frac{2}{t-1}+1\geq 2\sqrt{2}+1$$ 
$\star$ If $-\sqrt{2} <=t<-1 $ or $-1<t <1$ . Then using A.M G.M , $$\displaystyle -y = (1-t)+\frac{2}{1-t}>= 2\sqrt{2}-1$$
So $$y \geq 2\sqrt{2}-1$$ when $t=1-\sqrt{2}$ 

