Question 1126750
.


                The solution by the tutor @MathLover1 is not exactly correct,


                so I came to fix it.



<pre>
Start from this basic identity


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


We are given that


    3*sin(a) - 4*sin^3(a) = 2*sin(a).


It implies


    {{{(4*sin^2(a) - 1)*sin(a)}}} = 0.


So, either  sin(a) = 0   or   {{{4*sin^2(a) - 1}}} = 0;  the last is equivalent to  {{{sin^2(a)}}} = {{{1/4}}}.


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


Case 2.  If  {{{sin^2(a)}}} = {{{1/4}}},  then  cos(2a) = {{{1 - 2*sin^2(a))}}} = {{{1 - 2*(1/4)}}} = {{{1/2}}}.


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

Solved.