Question

If cosθ + cos^2 θ = 1,
then
sin^12 θ + 3 sin^10 θ + 3 sin^8 θ+ sin^6 θ + 2 sin^4 θ + 2 sin^2 θ – 2 =?

Answer 1

since sin^2 + cos^2 = 1,
sin^2 θ = cos θ

since cosθ + cos^2 θ = 1,
cosθ = ±(√5-1)/2

Just plug that in for sin^2 θ, and you have

((√5-1)/2)^6 + 3((√5-1)/2)^5 + 3((√5-1)/2)^4 + ((√5-1)/2)^3 + 2((√5-1)/2)^2 + 2((√5-1)/2) - 2
= 1
Thank you wolframalpha!

Now, how can we get that for ourselves?

If you let x = sin^2 θ, the above is

x^3 (x+1)^3 + 2(x^2+x-1)
= ((√5-1)/2)^3 ((√5+1)/2)^3 + 2(x^2+x-1)

But, since (√5-1)/2 is a root of x^2+x-1 = 0 (from our original condition),

= (5-1)^3/64 + 2(0)
= 1