Cosine of Integer Multiple of Argument/Formulation 2/Examples/Cosine of Sextuple Angle

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Example of Use of Cosine of Integer Multiple of Argument/Formulation 2

$\map \cos {6 \theta } = \cos^6 \theta \paren {1 - 15 \tan^2 \theta + 15 \tan^4 \theta - \tan^6 \theta}$


Proof

Follows directly from the Cosine of Integer Multiple of Argument/Formulation 2:

\(\ds \map \cos {6 \theta}\) \(=\) \(\ds \cos^n \theta \paren {1 - \dbinom n 2 \paren {\tan \theta}^2 + \dbinom n 4 \paren {\tan \theta}^4 - \cdots}\) Cosine of Integer Multiple of Argument/Formulation 2
\(\ds \) \(=\) \(\ds \cos^6 \theta \paren {1 - \dbinom 6 2 \paren {\tan \theta}^2 + \dbinom 6 4 \paren {\tan \theta}^4 - \dbinom 6 6 \paren {\tan \theta}^6}\)
\(\ds \) \(=\) \(\ds \cos^6 \theta \paren {1 - 15 \tan^2 \theta + 15 \tan^4 \theta - \tan^6 \theta}\)

$\blacksquare$