Cosine of Integer Multiple of Argument/Formulation 6

Theorem

For $n \in \Z$:

$\map \cos {n \theta} = \paren {-2 \sin \theta} \map \sin {\paren {n - 1} \theta} + \map \cos {\paren {n - 2} \theta}$

Proof

\(\ds \map \cos {n \theta}\)

\(=\)

\(\ds \map \cos {\paren {n - 1} \theta + \theta}\)

\(\ds \)

\(=\)

\(\ds -\sin \theta \map \sin {\paren {n - 1} \theta} + \cos \theta \map \cos {\paren {n - 1} \theta}\)

Cosine of Sum

\(\ds \)

\(=\)

\(\ds - \sin \theta \map \sin {\paren {n - 1} \theta} + \cos \theta \paren {-\sin \theta \map \sin {\paren {n - 2} \theta} + \cos \theta \map \cos {\paren {n - 2} \theta} }\)

Cosine of Sum

\(\ds \)

\(=\)

\(\ds -\sin \theta \map \sin {\paren {n - 1} \theta} - \sin \theta \cos \theta \map \sin {\paren {n - 2} \theta} + \cos^2 \theta \map \cos {\paren {n - 2} \theta}\)

\(\ds \)

\(=\)

\(\ds -\sin \theta \map \sin {\paren {n - 1} \theta} - \sin \theta \cos \theta \map \sin {\paren {n - 2} \theta} + \paren {1 - \sin^2 \theta } \map \cos {\paren {n - 2} \theta}\)

Sum of Squares of Sine and Cosine

\(\ds \)

\(=\)

\(\ds -\sin \theta \map \sin {\paren {n - 1} \theta} - \sin \theta \paren {\cos \theta \map \sin {\paren {n - 2} \theta} + \sin \theta \map \cos {\paren {n - 2} \theta} } + \map \cos {\paren {n - 2} \theta}\)

\(\ds \)

\(=\)

\(\ds -\sin \theta \map \sin {\paren {n - 1} \theta} - \sin \theta \map \sin {\paren {n - 1} \theta} + \map \cos {\paren {n - 2} \theta}\)

Sine of Sum

\(\ds \)

\(=\)

\(\ds \paren {-2 \sin \theta} \map \sin {\paren {n - 1} \theta} + \map \cos {\paren {n - 2} \theta}\)

$\blacksquare$