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If $a \cos \theta + b \sin \theta = m$ and $a \sin \theta - b \cos \theta = n$, then prove that $a^2 + b^2 = m^2 + n^2$.
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$m^2 + n^2 = (a \cos \theta + b \sin \theta)^2 + (a \sin \theta - b \cos \theta)^2$
$= a^2(\cos^2\theta + \sin^2\theta) + b^2(\sin^2 \theta + \cos^2 \theta)$
$= a^2 + b^2$
$= a^2(\cos^2\theta + \sin^2\theta) + b^2(\sin^2 \theta + \cos^2 \theta)$
$= a^2 + b^2$