If $\mathrm{\alpha },\mathrm{\beta }$ and $\gamma$ are three distinct real values such that $\frac{\sin \mathrm{\alpha }+\sin \mathrm{\beta }+\sin \mathrm{\gamma }}{\sin (\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma })}=\frac{\cos \mathrm{\alpha }+\cos \mathrm{\beta }+\cos \mathrm{\gamma }}{\cos (\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma })}=2$ then  $\cos (\mathrm{\alpha }+\mathrm{\beta })+\cos (\mathrm{\beta }+\mathrm{\gamma })+\cos (\mathrm{\gamma }+\mathrm{\alpha })=$___

put , a=cis $\alpha$,b=cis$\beta$, c=cis$\gamma$
$\Rightarrow \mathrm{a}+\mathrm{b}+\mathrm{c}=2\mathrm{abc}\Rightarrow \sum _{ }\frac{1}{\mathrm{ab}}=2$
$\Rightarrow \sum _{ }\cos \left(\alpha +\beta \right)=2$
$\frac{\sin \mathrm{\alpha }+\sin \mathrm{\beta }+\sin \mathrm{\gamma }}{\sin (\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma })}=\frac{\cos \mathrm{\alpha }+\cos \mathrm{\beta }+\cos \mathrm{\gamma }}{\cos (\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma })}$
$\frac{\sin \left(\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }\right)\left(\mathrm{sin\alpha }+\sin \mathrm{\beta }+\sin \mathrm{\gamma }\right)+\cos \left(\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }\right)\left(\mathrm{cos\alpha }+\cos \mathrm{\beta }+\cos \mathrm{\gamma }\right)}{\sin (\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma })\sin \left(\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }\right)+\cos \left(\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }\right)\cos \left(\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }\right)}$
$\left(\therefore \frac{\mathrm{a}}{\mathrm{b}}=\frac{\mathrm{c}}{\mathrm{d}}=\frac{\mathrm{xa}+\mathrm{yc}}{\mathrm{xb}+\mathrm{yd}}\right)$ (By ratio and proportion)
$\therefore \frac{\cos \left(\mathrm{\alpha }+\mathrm{\beta }\right)+\cos \left(\mathrm{\beta }+\mathrm{\gamma }\right)+\cos \left(\mathrm{\gamma }+\mathrm{\alpha }\right)}{1}=2$