Prove that cos^2A+cos^2(A+120)+cos^2(A-120)=3/2

${\cos ^2}2{\rm{A}} + {\cos ^2}2\left( {{\rm{A}} + {{120}^ \circ }} \right) + {\cos ^2}2\left( {{\rm{A}} - {{120}^ \circ }} \right) = {3 \over 2}$

We know that

${\cos ^2}{\rm{A}} = {{1 + \cos 2{\rm{A}}} \over 2}$                                    ..............(i)

${\cos ^2}2\left( {{\rm{A}} + {{120}^ \circ }} \right) = {{1 + \cos \left( {{\rm{2A}} + {{240}^ \circ }} \right)} \over 2}$         ..............(ii)

And   ${\cos ^2}2\left( {{\rm{A}} - {{120}^ \circ }} \right) = {{1 + \cos \left( {{\rm{2A}} - {{240}^ \circ }} \right)} \over 2}$..............(iii)

Adding all these three equations,

${\cos ^2}2{\rm{A}} + {\cos ^2}2\left( {{\rm{A}} + {{120}^ \circ }} \right) + {\cos ^2}2\left( {{\rm{A}} - {{120}^ \circ }} \right)$ = ${3 \over 2} + {{1 + \cos 2{\rm{A}}} \over 2} + {{1 + \cos \left( {{\rm{2A + }}{{240}^ \circ }} \right)} \over 2} + {{1 + \cos \left( {{\rm{2A}} - {{240}^ \circ }} \right)} \over 2}$

= ${{3 + 1 + \cos 2{\rm{A + }}1 + \cos \left( {{\rm{2A + }}{{240}^ \circ }} \right) + 1 + \cos \left( {{\rm{2A}} - {{240}^ \circ }} \right)} \over 2}$

= ${{3 + 3 + \cos 2{\rm{A}} + \cos \left( {{\rm{2A + }}{{240}^ \circ }} \right) + \cos \left( {{\rm{2A}} - {{240}^ \circ }} \right)} \over 2}$

= ${{3 + 3 + \cos 2{\rm{A}} + 2\cos 2{\rm{A}}.\cos {{240}^ \circ }} \over 2}$

= ${{3 + 3 + \cos 2{\rm{A}} - \cos 2{\rm{A}}} \over 2}$

= ${6 \over 2}$

$\Rightarrow$ ${\cos ^2}2{\rm{A}} + {\cos ^2}2\left( {{\rm{A}} + {{120}^ \circ }} \right) + {\cos ^2}2\left( {{\rm{A}} - {{120}^ \circ }} \right) = {3 \over 2}$