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If $\sin ^{2} \theta+\sin \theta=1$, then find $\cos ^{4} \theta+\cos ^{2} \theta$
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We have that $\sin ^{2} \theta+\sin \theta=1$. Subtracting $\sin ^{2} \theta$ from both sides and rearranging gives $\sin \theta=1-\sin ^{2} \theta=\cos ^{2} \theta$. Then $\cos ^{4} \theta+\cos ^{2} \theta=\cos ^{2} \theta\left(\cos ^{2} \theta+1\right)=\sin \theta(\sin \theta+1)=\sin ^{2} \theta+\sin \theta=1$.

by Diamond (71,587 points)

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