To rewrite the expression \( 8 - 8(\cos \theta - \sin \theta)^2 \) using double-angle formulas, we start by simplifying \( (\cos \theta - \sin \theta)^2 \).
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Expand \( (\cos \theta - \sin \theta)^2 \): \[ (\cos \theta - \sin \theta)^2 = \cos^2 \theta - 2\cos \theta \sin \theta + \sin^2 \theta \] Using the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \), we can substitute this in: \[ \cos^2 \theta + \sin^2 \theta = 1 \implies (\cos \theta - \sin \theta)^2 = 1 - 2\cos \theta \sin \theta \]
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Substitute this back into the original expression: \[ 8 - 8(\cos \theta - \sin \theta)^2 = 8 - 8(1 - 2\cos \theta \sin \theta) \] Simplifying inside the parentheses: \[ = 8 - 8 + 16\cos \theta \sin \theta = 16\cos \theta \sin \theta \]
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Now, we can use the double-angle identity for sine, which states: \[ \sin(2\theta) = 2\sin \theta \cos \theta \implies \cos \theta \sin \theta = \frac{1}{2} \sin(2\theta) \]
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Substitute this back into our expression: \[ 16\cos \theta \sin \theta = 16 \cdot \frac{1}{2} \sin(2\theta) = 8\sin(2\theta) \]
Therefore, the expression \( 8 - 8(\cos \theta - \sin \theta)^2 \) can be rewritten as:
\[ \boxed{8\sin(2\theta)} \]