Squares of Linear Combination of Sine and Cosine

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Theorem

$\left({a \cos x + b \sin x}\right)^2 + \left({b \cos x - a \sin x}\right)^2 = a^2 + b^2$

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \left({a \cos x + b \sin x}\right)^2 + \left({b \cos x - a \sin x}\right)^2$	$=$	$\displaystyle a^2 \cos^2 x + 2 a b \cos x \ \sin x + b^2 \sin^2 x$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $		$\displaystyle \quad + \ b^2 \cos^2 x - 2 a b \sin x \ \cos x + a^2 \sin^2 x$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \left({a^2 + b^2}\right) \left({\sin^2 x + \cos^2 x}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle a^2 + b^2$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Sum of Squares of Sine and Cosine

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left({a \cos x + b \sin x}\right)^2 + \left({b \cos x - a \sin x}\right)^2\)	\(=\)	\(\displaystyle a^2 \cos^2 x + 2 a b \cos x \ \sin x + b^2 \sin^2 x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\)	\(\displaystyle \quad + \ b^2 \cos^2 x - 2 a b \sin x \ \cos x + a^2 \sin^2 x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({a^2 + b^2}\right) \left({\sin^2 x + \cos^2 x}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle a^2 + b^2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Sum of Squares of Sine and Cosine