If the minimum value of $\mid \sin \mathrm{x}+\cos \mathrm{x}+\tan \mathrm{x}+\cot x+\sec \mathrm{x}+\mathrm{cosec}\mathrm{x}\mid ,\mathrm{x}\in \mathrm{R}$ is $\sqrt{\mathrm{m}}+\mathrm{n}$ then m + n equal $(\mathrm{m},\mathrm{n}\in \mathrm{z})$

Playing a little bit with the expression one discovers that if 

we get $\mathrm{t}=\sin \mathrm{x}+\cos \mathrm{x},\frac{{\mathrm{t}}^2-1}{2}=\mathrm{sinx} \mathrm{cosx}$,then given expression $=\left|\mathrm{t}+\frac{1}{\frac{{\mathrm{t}}^2-1}{2}}+\frac{\mathrm{t}}{\frac{{\mathrm{t}}^2-1}{2}}\right|$

$\begin{array}{l}=\left|\mathrm{t}+\frac{2}{{\mathrm{t}}^2-1}+\frac{2\mathrm{t}}{{\mathrm{t}}^2-1}\right|\\ \mathrm{y}=\left|\mathrm{t}+\frac{2}{\mathrm{t}-1}\right|=\left|\mathrm{t}-1+\frac{2}{\mathrm{t}-1}+1\right|\end{array}$

If $1<\mathrm{t}\le \sqrt{2}$ then $\mathrm{y}=\mathrm{t}-1+\frac{2}{\mathrm{t}-1}+1\ge 2\sqrt{2}+1$

If $-\sqrt{2}\le \mathrm{t}<-1$ then $\mathrm{y}=-1+1-\mathrm{t}+\frac{2}{1-\mathrm{t}}\ge 2\sqrt{2}-1$

$\begin{array}{l}\therefore {\mathrm{y}}_{min}=2\sqrt{2}-1=\sqrt{8}-1\\ \therefore \mathrm{m}+\mathrm{n}=7\end{array}$