Question 1142056
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First use the identity


    {{{cos(4pi/7)}}} = {{{-cos(3pi/7)}}},


which gives you


    {{{cos^2(4pi/7)}}} = {{{cos^2(3pi/7)}}}.


Therefore, the left side of the given hypothetical identity becomes


    {{{cos^2 (4pi/7)}}} - {{{sin^2 (3pi/7)}}} = {{{cos^2 (3pi/7)}}} - {{{sin^2 (3pi/7)}}}.    (1)


Next, use the trigonometric identity  


    cos(a)*cos(b) - sin(a)*sin(b) = cos(a+b).


It allows you to continue the line (1) in this way


    {{{cos^2 (4pi/7)}}} - {{{sin^2 (3pi/7)}}} = {{{cos^2 (3pi/7)}}} - {{{sin^2 (3pi/7)}}} = {{{cos(3pi/7 + 3pi/7)}}} = {{{cos(6pi/7)}}}.


Thus 


    {{{cos^2 (4pi/7)}}} - {{{sin^2 (3pi/7)}}} = {{{cos(6pi/7)}}}.


It is what has to be proved.
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