Let X be any angle.
sinX+(sinX)^2=1; (sinX)^2+sinX+1/4=5/4; (sinX+1/2)^2=5/4, sinX=-(1/2)+sqrt(5)/2.
(cosX)^4+(cosX)^2=1; (cosX)^2=-(1/2)+sqrt(5)/2. (sinX)^2=1-(cosX)^2=1-(-(1/2)+sqrt(5)/2)=(3/2)Tsqrt(5)/2.
To prove the equality: (-(1/2)+sqrt(5)/2)^2 must be equal to (3/2)Tsqrt(5)/2.
(1/4)(1+5T2sqrt(5))=(6/4)Tsqrt(5)/2=(3/2)Tsqrt(5)/2. So, because this is true, the equality must also be true.
Alternatively:
sinX=(cosX)^2; sinX=1-(sinX)^2; (sinX)^2+sinX=1. So the solution of the first equation is also the condition of sinX being equal to (cosX)^2. The two equations are therefore equivalent.