If $\cos \mathrm{\alpha } + \cos \mathrm{\beta } + \cos \mathrm{\gamma } = 0 = \sin \mathrm{\alpha } + \sin \mathrm{\beta } + \sin \mathrm{\gamma } ,$ 
then match the following

	LIst I		List II
I)	$\cos 3\mathrm{\alpha } + \cos 3\mathrm{\beta } + \cos 3\mathrm{\gamma } =$		0
II)	$\sin 2\mathrm{\alpha } + \sin 2\mathrm{\beta } + \sin 2\mathrm{\gamma } =$		3
III)	${\cos }^2 \mathrm{\alpha } + {\cos }^2 \mathrm{\beta }+ {\cos }^2 \mathrm{\gamma } =$		3/2
IV)	$\cos (2\mathrm{\alpha }-\mathrm{\beta }-\mathrm{\gamma }) + \cos (2\mathrm{\beta }-\mathrm{\gamma }-\mathrm{\alpha })$$+ \cos (2\mathrm{\gamma }-\mathrm{\alpha }-\mathrm{\beta }) =$		$3\cos (\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma })$

                                                                            

$\begin{array}{rcl}\mathrm{Let} \mathrm{a}& =& \mathrm{cis} \mathrm{\alpha }, \mathrm{b}=\mathrm{cis} \mathrm{\beta }, \mathrm{c}=\mathrm{cis} \mathrm{\gamma }\end{array}$

$\begin{array}{rcl}\cos \mathrm{\alpha }+\cos \mathrm{\beta }+\cos \mathrm{\gamma }& =& 0, \sin \mathrm{\alpha }+\sin \mathrm{\beta }+\sin \mathrm{\gamma }=0\end{array}$

$\begin{array}{rcl}\mathrm{a}+\mathrm{b}+\mathrm{c}& =& (\cos \mathrm{\alpha }+\cos \mathrm{\beta }+\cos \mathrm{\gamma })+\mathrm{i}(\sin \mathrm{\alpha }+\sin \mathrm{\beta }+\sin \mathrm{\gamma })\end{array}$

$\begin{array}{rcl}\mathrm{a}+\mathrm{b}+\mathrm{c}& =& 0\end{array}$

$\begin{array}{rcl} \frac{1}{\mathrm{a}}+\frac{1}{\mathrm{b}}+\frac{1}{\mathrm{c}}& =& \mathrm{i}(\cos \mathrm{\alpha }+\cos \mathrm{\beta }+\cos \mathrm{\gamma })-1(\sin \mathrm{\alpha }+\sin \mathrm{\beta }+\sin \mathrm{\gamma })\end{array}$

$\begin{array}{rcl}\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{b}}+\frac{1}{\mathrm{c}}& =& 0 \Rightarrow \mathrm{ab}+\mathrm{bc}+\mathrm{ca}=0\end{array}$

$\begin{array}{rcl}{(\mathrm{a}+\mathrm{b}+\mathrm{c})}^2& =& \Rightarrow {\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2+2(\mathrm{ab}+\mathrm{bc}+\mathrm{ca})=0\end{array}$

$\begin{array}{rcl} & \Rightarrow & {\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2=0\end{array}$

$\begin{array}{rcl} & & {(\cos \mathrm{\alpha }+\mathrm{i} \sin \mathrm{\alpha })}^2+{(\cos \mathrm{\beta }+\mathrm{i} \sin \mathrm{\beta })}^2+{(\cos \mathrm{\gamma }+\mathrm{i} \sin \mathrm{\gamma })}^2=0\end{array}$

$\begin{array}{rcl} & \Rightarrow & \cos 2\mathrm{\alpha }+\mathrm{i} \sin 2\mathrm{\alpha }+\cos 2\mathrm{\beta }+\mathrm{i} \sin 2\mathrm{\beta }+\cos 2\mathrm{\gamma }+\mathrm{i} \sin 2\mathrm{\gamma }=0\end{array}$

$\begin{array}{rcl} & \Rightarrow & \cos 2\mathrm{\alpha }+\cos 2\mathrm{\beta }+\cos 2\mathrm{\gamma }=0, \sin 2\mathrm{\alpha }+\sin 2\mathrm{\beta }+\sin 2\mathrm{\gamma }=0\end{array}$

$\begin{array}{rcl}2{\cos }^2\mathrm{\alpha }-1+2{\cos }^2\mathrm{\alpha }-2+2{\cos }^2\mathrm{\gamma }-1& =& 0\end{array}$

$\begin{array}{rcl} & \Rightarrow & {\cos }^2\mathrm{\alpha }+{\cos }^2\mathrm{\beta }+{\cos }^2\mathrm{\gamma }=3/2\end{array}$

$\begin{array}{rcl}\mathrm{a}+\mathrm{b}+\mathrm{c}& =& 0 \Rightarrow {\mathrm{a}}^3+{\mathrm{b}}^3+{\mathrm{c}}^3=3\mathrm{abc}\end{array}$

$\begin{array}{rcl}{(\cos \mathrm{\alpha }+\mathrm{i} \sin \mathrm{\alpha })}^3+{(\cos \mathrm{\beta }+\mathrm{i} \sin \mathrm{\beta })}^3+{(\cos \mathrm{\gamma }+\mathrm{i} \sin \mathrm{\gamma })}^3& =& 3 \mathrm{cis} \mathrm{\alpha } \mathrm{cis} \mathrm{\beta } \mathrm{cis} \mathrm{\gamma }\end{array}$

$\begin{array}{rcl} & \Rightarrow & \cos 3\mathrm{\alpha }+\cos 3\mathrm{\beta }+\cos 3\mathrm{\gamma }=3\cos (\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma })\end{array}$

$\begin{array}{rcl} & & \\ \cos 3\mathrm{\alpha }+\mathrm{i} \sin 3\mathrm{\alpha }+\cos 3\mathrm{\beta }+\mathrm{i} \sin 3\mathrm{\beta }+\cos 3\mathrm{\gamma }+\mathrm{i} \sin 3\mathrm{\gamma }& =& 3\mathrm{cis}(\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma })\\ & & (\cos 3\mathrm{\alpha }+\cos 3\mathrm{\beta }+\cos 3\mathrm{\gamma })+\mathrm{i}(\sin 3\mathrm{\alpha }+\sin 3\mathrm{\beta }+\sin 3\mathrm{\gamma })\\ & =& 3\left(\cos \right(\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma })+\mathrm{i} \sin (\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }\left)\right)\end{array}$

$\begin{array}{rcl}\mathrm{a}+\mathrm{b}+\mathrm{c}& =& 0 \Rightarrow {\mathrm{a}}^3+{\mathrm{b}}^3+{\mathrm{c}}^3=3\mathrm{abc}\end{array}$

$\begin{array}{rcl}\frac{{\mathrm{a}}^2}{\mathrm{bc}}+\frac{{\mathrm{b}}^2}{\mathrm{ca}}+\frac{{\mathrm{c}}^2}{\mathrm{ab}}& =& 3\end{array}$

$\begin{array}{rcl}\frac{{(\mathrm{cis} \mathrm{\alpha })}^2}{\mathrm{cis} \mathrm{\beta } \mathrm{cis} \mathrm{\gamma }}+\frac{{(\mathrm{cis} \mathrm{\beta })}^2}{\mathrm{cis} \mathrm{\gamma } \mathrm{cis} \mathrm{\alpha }}+\frac{{(\mathrm{cis} \mathrm{\gamma })}^2}{\mathrm{cis} \mathrm{\alpha } \mathrm{cis} \mathrm{\beta }}& =& 3\end{array}$

$\begin{array}{rcl}\mathrm{cis}(2\mathrm{\alpha }-\mathrm{\beta }-\mathrm{\gamma })+\mathrm{cis}(2\mathrm{\beta }-\mathrm{\gamma }-\mathrm{\alpha })+\mathrm{cis}(2\mathrm{\alpha }-\mathrm{\alpha }-\mathrm{\beta })& =& 3\end{array}$

$\begin{array}{rcl} & \Rightarrow & \cos (2-\mathrm{\alpha }-\mathrm{\beta }-\mathrm{\gamma })+\cos (2\mathrm{\beta }-\mathrm{\gamma }-\mathrm{\alpha })+\cos (2\mathrm{\gamma }-\mathrm{\alpha }-\mathrm{\beta })=3\end{array}$