<s>[INST] Trigonometry - Finding value of expression Question: Given $\sin x + \sin^2 x = 1$ then find the value of $$\cos^{12} x + 3\cos^{10} x + 3\cos^8 x + 6\cos^6 x + 2\cos^4 x + \cos^2 x -2$$
I have no idea where to start on this question. Please help me!
 [/INST] HINT:
$$\sin x=1-\sin^2x=\cos^2x$$
Squaring we get $$\sin^2x=(\cos^2x)^2\iff 1-c^2=c^4\iff c^4+c^2-1=0$$ where $c=\cos x$
Now  establish that $c^4+c^2-1$ is a factor of $$c^{12}   + 3c^{10}   + 3c^8  + 6c^6   + 2c^4   + c^2   -2$$  
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