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

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

$\map \cos {5 \theta } = \cos^5 \theta \paren {1 - 10 \tan^2 \theta + 5 \tan^4 \theta}$

Proof

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

\(\ds \map \cos {5 \theta}\)

\(=\)

\(\ds \cos^n \theta \paren {1 - \dbinom n 2 \paren {\tan \theta}^2 + \dbinom n 4 \paren {\tan \theta}^4 - \cdots}\)

Cosine of Integer Multiple of Argument: Formulation 2

\(\ds \)

\(=\)

\(\ds \cos^n \theta \paren {1 - \dbinom 5 2 \paren {\tan \theta}^2 + \dbinom 5 4 \paren {\tan \theta}^4 - \cdots}\)

\(\ds \)

\(=\)

\(\ds \cos^5 \theta \paren {1 - 10 \tan^2 \theta + 5 \tan^4 \theta}\)

$\blacksquare$