sin^2(x)=1-sin(x). Quadratic in sin(x): sin^2(x)+sin(x)-1=0: sin(x)=(-1±√5)/2. This solution is substituted later into the solution of the main problem (see end).
cos^2(x)=1-sin^2(x)=1-(1-sin(x))=sin(x).
cos^12(x)=(cos^2(x))^6=sin^6(x); cos^10(x)=sin^5(x); cos^8(x)=sin^4(x); cos^6(x)=sin^3(x).
sin^3(x)=(1-sin(x))sin(x)=sin(x)-sin^2(x)=sin(x)-1+sin(x)=2sin(x)-1.
The expression cos^12(x)+cos^10(x)+cos^8(x)+cos^6(x) becomes:
sin^6(x)+sin^5(x)+sin^4(x)+sin^3(x)=
sin^3(x)(sin^3(x)+sin^2(x)+sin(x)+1)=(2sin(x)-1)(2sin(x)-1+1-sin(x)+sin(x)+1)=
(2sin(x)-1)(2sin(x)+1)=4sin^2(x)-1=4-4sin(x)-1=3-4sin(x)=5±2√5 (see solution of quadratic earlier) =9.4721 or 0.5279 (roots of X^2-10X+5=0).