The minimum value of the expression $\left|\sin x+\cos x+\tan x+\cot x+\sec x+\mathrm{cosec} x\right|$ can be expressed as $(\sqrt a-\sqrt b)$ where a and b are natural number then find the value of $(a-b)?$
My attempt:I applied AM-GM inequality (but AM GM inequality can be applied when all quantities are positive and all trigonometric ratios are not positive except when in first quadrant.)
$$\frac{\sin x+\cos x+\tan x+\cot x+\sec x+\mathrm{cosec} x}{6}\geq\sqrt[6]{\sin x.\cos x.\tan x.\cot x.\sec x.\mathrm{cosec} x}$$
$\sin x+\cos x+\tan x+\cot x+\sec x+\mathrm{cosec} x\geq6 $
but this is the minimum value of $\sin x+\cos x+\tan x+\cot x+\sec x+\mathrm{cosec}x$ ,not minimum value of $\left|\sin x+\cos x+\tan x+\cot x+\sec x+\mathrm{cosec} x\right|$ and it not expressible as $\sqrt a -\sqrt b$
Someone help me find $a$ and $b$.Thanks in advance..