Squares of Linear Combination of Sine and Cosine
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Theorem
- $\paren {a \cos x + b \sin x}^2 + \paren {b \cos x - a \sin x}^2 = a^2 + b^2$
Proof
| \(\ds \paren {a \cos x + b \sin x}^2 + \paren {b \cos x - a \sin x}^2\) | \(=\) | \(\ds a^2 \cos^2 x + 2 a b \cos x \ \sin x + b^2 \sin^2 x\) | ||||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds b^2 \cos^2 x - 2 a b \sin x \ \cos x + a^2 \sin^2 x\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {a^2 + b^2} \paren {\sin^2 x + \cos^2 x}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds a^2 + b^2\) | Sum of Squares of Sine and Cosine |
$\blacksquare$