Posted by Anonymous on Friday, January 11, 2013 at 10:49am.
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
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