Question 1086313
.
{{{graph( 330, 330, -0.5, 6.5, -1.5, 2.5,
          sin(x) + (sin(x))*(sin(x)), 1 
)}}}


Plots y = {{{sin(x) + sin^2(x)}}} (red) and y = 1 (green)



<pre>
1.  {{{sin(x) + sin^2(x)}}} = 1  ====>  {{{sin^2(x) + sin(x) -1}}} = 0  =====>

    solve this quadratic equation for sin(x) to get a single value for sin(x) = {{{(-1 + sqrt(5))/2}}}.


2.  {{{sin(x) + sin^2(x)}}} = 1  ====>  sin(x) = {{{1 - sin^2(x)}}} = {{{cos^2(x)}}}.

    Thus {{{cos^2(x)}}} = {{{(-1 + sqrt(5))/2}}}.


3.  {{{cos^12(x) + 3cos^10(x) + 3cos^8(x) + cos^6(x)}}} = {{{cos^6(x)*(cos^6(x) + 3*cos^4(x) + 3*cos^2(x) + 1)}}} = {{{cos^6(x)*(cos^2(x)+1)^3}}} = 

    = {{{((-1 + sqrt(5))/2)^3}}} . {{{((1 + sqrt(5))/2)^3}}} = {{{((5-1)/4)^3}}} = 1.


4.  ====>  {{{cos^12(x) + 3cos^10(x) + 3cos^8(x) + cos^6(x) - 2}}} = 1 - 2 = -1.
</pre>


<U>Answer</U>.   -1.


Solved.