Question 1126751
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Start from this basic identity


    cos(3a)=4*cos^3(a) - 3*cos(a).


Then 2*cos(3a) = 8*cos^3(a) - 6*cos(a),  and we are given that


    8*cos^3(a) - 6*cos(a) = cos(a).


It implies


    {{{(8cos^2(a) - 7)*cos(a)}}} = 0.


So, either  cos(a) = 0   or   {{{8cos^2(a) - 7}}} = 0;  the last is equivalent to  {{{cos^2(a)}}} = {{{7/8}}}.


Case 1.  If  cos(a) = 0,  then  cos(2a) = {{{2*cos^2(a) -1}}} = -1.


Case 2.  If  {{{cos^2(a)}}} = {{{7/8}}},  then  cos(2a) = {{{2*cos^2(a) - 1}}} = {{{2*(7/8)-1}}} = {{{14/8-1}}} = {{{6/8}}} = {{{3/4}}}.


<U>Answer</U>.  If  2*cos(3a) = cos(a),  then EITHER  cos(2a) = -1  OR  cos(2a) = {{{3/4}}}.
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Solved.