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If $\sin \theta=\cos ^{2} \theta$, then what is $\cos ^{2} \theta\left(1+\cos ^{2} \theta\right)$ equal to?
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The correct answer is:
1
Let $\sin \theta=\cos ^{2} \theta$
$\Rightarrow \sin ^{2} \theta=\cos ^{4} \theta$
Consider $\cos ^{2} \theta\left(1+\cos ^{2} \theta\right)=\cos ^{2} \theta+\cos ^{4} \theta$
$=\cos ^{2} \theta+\sin ^{2} \theta \quad$ (using 1)
$=1$
$\Rightarrow \sin ^{2} \theta=\cos ^{4} \theta$
Consider $\cos ^{2} \theta\left(1+\cos ^{2} \theta\right)=\cos ^{2} \theta+\cos ^{4} \theta$
$=\cos ^{2} \theta+\sin ^{2} \theta \quad$ (using 1)
$=1$
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