Secrets of Chapel Dulcinea: You Won’t Believe What Lies Inside This Haunting Shrine!
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Solution: We begin by rewriting $ f(\theta) $ using trigonometric identities. Observe:
f(\theta) = \sin^4 \theta - 4\sin^2 \theta \cos^2 \theta + \cos^4 \theta.
Group the first and last terms:
f(\theta) = (\sin^4 \theta + \cos^4 \theta) - 4\sin^2 \theta \cos^2 \theta.
Recall that:
\sin^4 \theta + \cos^4 \theta = (\sin^2 \theta + \cos^2 \theta)^2 - 2\sin^2 \theta \cos^2 \theta = 1 - 2\sin^2 \theta \cos^2 \theta.
Substitute into $ f(\theta) $:
f(\theta) = (1 - 2\sin^2 \theta \cos^2 \theta) - 4\sin^2 \theta \cos^2 \theta = 1 - 6\sin^2 \theta \cos^2 \theta.
Now use the identity $ \sin(2\theta) = 2\sin\theta \cos\theta $, so:
\sin^2(2\theta) = 4\sin^2\theta \cos^2\theta \Rightarrow \sin^2\theta \cos^2\theta = \frac{1}{4} \sin^2(2\theta).