Cosine of Integer Multiple of Argument/Formulation 3
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Theorem
For $n \in \Z_{>0}$:
| \(\ds \cos n \theta\) | \(=\) | \(\ds \cos \paren {n - 1} \theta \cos \theta + \paren {1 - \sec^2 \theta} \cos^n \theta \paren {1 + 1 + \frac {\cos 2 \theta} {\cos^2 \theta} + \frac {\cos 3 \theta} {\cos^3 \theta} + \cdots + \frac {\cos \paren {n - 2} \theta} {\cos^{n - 2} \theta} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \cos \paren {n - 1} \theta \cos \theta + \paren {1 - \sec^2 \theta} \cos^n \theta \sum_{k \mathop = 0}^{n - 2} \frac {\cos k \theta} {\cos^k \theta}\) |
Proof
| \(\ds \cos n \theta\) | \(=\) | \(\ds \map \cos {\paren {n - 1} \theta + \theta}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \cos \paren {n - 1} \theta \cos \theta - \sin \paren {n - 1} \theta \sin \theta\) | Cosine of Sum | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos \paren {n - 1} \theta \cos \theta - \paren {\sin \theta \cos^{n - 2} \theta \sum_{k \mathop = 0}^{n - 2} \frac {\cos k \theta} {\cos^k \theta} } \sin \theta\) | Sine of Integer Multiple of Argument: Formulation 3 | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos \paren {n - 1} \theta \cos \theta - \paren {\sin^2 \theta \cos^{n - 2} \theta \sum_{k \mathop = 0}^{n - 2} \frac {\cos k \theta} {\cos^k \theta} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \cos \paren {n - 1} \theta \cos \theta - \paren {\paren {1 - \cos^2 \theta} \cos^{n - 2} \theta \sum_{k \mathop = 0}^{n - 2} \frac {\cos k \theta} {\cos^k \theta} }\) | Sum of Squares of Sine and Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos \paren {n - 1} \theta \cos \theta + \paren {\paren {\cos^2 \theta - 1} \cos^{n - 2} \theta \sum_{k \mathop = 0}^{n - 2} \frac {\cos k \theta} {\cos^k \theta} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \cos \paren {n - 1} \theta \cos \theta + \paren {1 - \sec^2 \theta} \cos^n \theta \sum_{k \mathop = 0}^{n - 2} \frac {\cos k \theta} {\cos^k \theta}\) | Secant is Reciprocal of Cosine |
$\blacksquare$