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If $\sin \theta + \sin^2 \theta = 1$, then prove that $\cos^2\theta + \cos^4 \theta = 1$.
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$$\begin{aligned}& \sin \theta + \sin^2 \theta = 1 \\ & \Rightarrow \sin \theta = 1 - \sin^2 \theta = \cos^2 \theta \\ & \therefore \cos^2 \theta + \cos^4 \theta = \cos^2 \theta (1 + \cos^2 \theta) \\ & = \sin \theta (1 + \sin \theta) \\ & = \sin \theta + \sin^2 \theta= 1\end{aligned}$$